Recent zbMATH articles in MSC 42 https://www.zbmath.org/atom/cc/42 2022-06-24T15:10:38.853281Z Unknown author Werkzeug Introduction to the special issue on harmonic analysis and machine learning https://www.zbmath.org/1485.00014 2022-06-24T15:10:38.853281Z (no abstract) Sharp refined quadratic estimations of Shafer's inequalities https://www.zbmath.org/1485.26018 2022-06-24T15:10:38.853281Z "Zhu, Ling" https://www.zbmath.org/authors/?q=ai:zhu.ling Summary: In this paper, using the power series expansions of $$(\tan x)^k$$ ($$k = 1, 2, 3$$) and the monotonicity of a function involving the Riemann's zeta function, we sharpen the quadratic estimations of Shafer's inequalities which is refined by \textit{Y. Nishizawa} [J. Inequal. Appl. 2017, Paper No. 40, 11 p. (2017; Zbl 1357.26042)]. The reverse Hölder inequality for an elementary function https://www.zbmath.org/1485.26025 2022-06-24T15:10:38.853281Z "Korenovskii, A. O." https://www.zbmath.org/authors/?q=ai:korenovskii.a-o Summary: For a positive function $$f$$ on the interval $$[0,1]$$, the power mean of order $$p\in\mathbb{R}$$ is defined by $\|\, f\,\|_p=\left(\int_0^1 f^p(x)\,dx\right)^{1/p}\quad(p\ne0),\quad\|\, f\,\|_0=\exp\left(\int_0^1\ln f(x)\,dx\right).$ Assume that $$0<A<B$$, $$0<\theta<1$$ and consider the step function $$g_{A<B,\theta}=B\cdot\chi_{[0,\theta)}+A\cdot\chi_{[\theta,1]}$$, where $$\chi_E$$ is the characteristic function of the set $$E$$. Let $$-\infty<p<q<+\infty$$. The main result of this work consists in finding the term $C_{p<q,A<B}=\max\limits_{0\le\theta\le1}\frac{\|\,g_{A<B,\theta}\,\|_q}{\|\,g_{A<B,\theta}\,\|_p}.$ For fixed $$p<q$$, we study the behaviour of $$C_{p<q,A<B}$$ and $$\theta_{p<q,A<B}$$ with respect to $$\beta=B/A\in(1,+\infty)$$. The cases $$p=0$$ or $$q=0$$ are considered separately. The results of this work can be used in the study of the extremal properties of classes of functions, which satisfy the inverse Hölder inequality, e.g. the Muckenhoupt and Gehring ones. For functions from the Gurov-Reshetnyak classes, a similar problem has been investigated in [the author, Ann. Mat. Pura Appl. (4) 195, No. 2, 659--680 (2016; Zbl 1342.26044)]. Spectrality of Sierpinski-type self-affine measures https://www.zbmath.org/1485.28016 2022-06-24T15:10:38.853281Z "Lu, Zheng-Yi" https://www.zbmath.org/authors/?q=ai:lu.zhengyi "Dong, Xin-Han" https://www.zbmath.org/authors/?q=ai:dong.xinhan "Liu, Zong-Sheng" https://www.zbmath.org/authors/?q=ai:liu.zongsheng The paper deals with the concept of spectral measures on fractal graphs, mainly the Sierpinski triangle. The authors tried to apply IFS to construct spectral measures on this special graph. The basic idea in the present paper reposes on the use of the projection of the spectrum on one coordinate (the first for example) to get necessary and sufficient conditions for the construction of a spectral measure associated to the IFS. This is interesting as it shows that the spectrality of the measure may be only dependent of the first eigenvalue instead of the whole spectrum. Besides, the paper joins a series of fascinating results on spectral theory, mainly fractal measures, and also provides nice mathematical techniques that may be applied in other related and similar problems such as n-dimensional fractals. Reviewer: Anouar Ben Mabrouk (Monastir) A Rudin-de Leeuw type theorem for functions with spectral gaps https://www.zbmath.org/1485.30016 2022-06-24T15:10:38.853281Z "Dyakonov, Konstantin M." https://www.zbmath.org/authors/?q=ai:dyakonov.konstantin-m The goal of this paper is to establish a generalization of the Rudin-de Leeuw theorem on the extreme points of the unit ball of the Hardy space $$H^1$$ consisting of analytic functions in the unit disk whose boundary functions have finite $$L^1$$-norm on the boundary $$\mathbb{T}$$ of the unit disk; that is, $\|f\|_1=\frac{1}{2\pi}\int_{\mathbb{T}}|f(z)| |dz|<\infty.$ It is well known that $$H^1$$ equals functions $$f\in L^1(\mathbb{T})$$ with $$\hat{f}(k)=0$$ for negative integers $$k$$; here $$\hat{f}(k)$$ stands for the Fourier coefficients. Using the notation $\operatorname{spec} f =\{k\in\mathbb{Z}: \hat{f}(k)\neq 0\},$ we may write $H^1=\{f\in L^1: \operatorname{spec} f\subset \mathbb{Z}_+\},$ where $$\mathbb{Z}_+$$ is the set of non-negative integers.\par Recall that if $$X$$ is a normed space, the closed unit ball of $$X$$ is denoted by $$\operatorname{ball}(X)$$; moreover, an element $$x\in\operatorname{ball}(X)$$ is called an \emph{extreme point} if it is not an interior point of any line segment contained in $$\operatorname{ball}(X)$$. Note that any such point is a unit-norm vector. \par \textit{K. de Leeuw} and \textit{W. Rudin} [Pac. J. Math. 8, 467--485 (1958; Zbl 0084.27503)] proved that a unit-norm function $$f\in H^1$$ is an extreme point of $$\operatorname{ball}(H^1)$$ if and only if it is an outer function (see also [\textit{J. B. Garnett}, Bounded analytic functions. Pure and Applied Mathematics, 96. New York etc.: Academic Press, A subsidiary of Harcourt Brace Javanovich, Publishers. (1981; Zbl 0469.30024)] and [\textit{K. Hoffman}, Banach spaces of analytic functions. Reprint of the 1962 original. New York: Dover Publications, Inc. (1988; Zbl 0734.46033)]). The author of the present paper considers a finite set of positive integers $\mathcal{K}=\{k_1,\ldots,k_M\},$ and defines the subspace $H^1_{\mathcal{K}}:=\{f\in H^1: \operatorname{spec} f\subset \mathbb{Z}_+\setminus \mathcal{K}\}.$ The purpose of this paper is to characterize the extreme points of $$\operatorname{ball}(H^1_{\mathcal{K}})$$ equipped with $$L^1$$ norm. The main result of this paper (Theorem 1) states that a unit-norm function $$f\in\operatorname{ball}(H^1_{\mathcal{K}})$$ with inner-outer factorization $$f=IF$$ is an extreme point if and only if $$I$$ is a finite Blaschke product whose degree $$m$$ does not exceed $$M$$; moreover, a certain block matrix (built by using the outer function $$F$$ and the $$m$$ zeros of $$I$$) has finite rank. Reviewer: Ali Abkar (Qazvin) Holomorphic function spaces on homogeneous Siegel domains https://www.zbmath.org/1485.32003 2022-06-24T15:10:38.853281Z "Calzi, Mattia" https://www.zbmath.org/authors/?q=ai:calzi.mattia "Peloso, Marco M." https://www.zbmath.org/authors/?q=ai:peloso.marco-maria The authors study holomorphic functions on homogeneous Siegel domains. They concentrate mainly on weighted mixed norm Bergman spaces. The problems considered include: sampling, atomic decomposition, duality, boundary values, and boundedness of the Bergman projectors. The work consists of an introduction and five chapters, it also contains an appendix devoted to mixed norm spaces. We shall briefly describe the contents of the consecutive chapters. In the introduction, the authors motivate their study by discussing the simplest example of a Siegel domain: the upper half-plane $$\mathbb{C}_{+}:=\mathbb{R}+i\mathbb{R}_{+}^{*}$$. They also present known results concerning function theory on Siegel domains. Let $$E$$ be a vector space over $$\mathbb{C}$$ of finite dimension $$n$$ and $$F$$ be a vector space over $$\mathbb{R}$$ of finite dimension $$m>0$$. Let $$\Omega$$ be an open convex cone with vertex $$0$$ in $$F$$ and not containing any affine lines. Also, let $$\Phi\colon E\times E\rightarrow F_{\mathbb{C}}$$, $$F_{\mathbb{C}}$$ is the complexification of $$F$$, i.e., $$F_{\mathbb{C}}=F\otimes_{\mathbb{R}}\mathbb{C}$$, be a positive non-degenerate hermitian mapping, that is: \begin{itemize} \item[(i)] $$\Phi$$ is linear in the first argument; \item[(ii)] $$\Phi(\zeta,\zeta^{'})=\overline{\Phi(\zeta^{'},\zeta)}$$ for all $$\zeta,\zeta^{'}\in E$$; \item[(iii)] $$\Phi$$ is non-degenerate; \item[(iv)] $$\Phi(\zeta):=\Phi(\zeta,\zeta)\in \overline{\Omega}$$. \end{itemize} {\em The Siegel domain of type II} associated with the cone $$\Omega$$ and the mapping $$\Phi$$ is defined the following way: $D:=\{(\zeta,z)\in E\times F_{\mathbb{C}}\colon \Im z-\Phi(\zeta)\in \Omega\}.$ Chapter one contains, beside the above definition and basic examples, definitions of the Fourier transform, the Bergman and the Hardy spaces, the formulation of the corresponding Paley-Wiener theorems and a discussion of the Kohn Laplacian. In Chapter two the authors introduce various objects related to homogeneous Siegel domains of type II. This includes a discussion of $$T$$-algebras and the associated homogeneous cones, the generalized power functions $$\Delta_{\Omega}^{s}$$, $$\Delta_{\Omega^{'}}^{s}$$, $$\Omega^{'}$$ the dual cone and the associated gamma and beta functions. They also introduce the corresponding Bergman metric. Chapter three is devoted to the study of the weighted Bergman spaces $$A_{s}^{p,q}(D)$$. These spaces are defined for $$\mathbf{s}\in\mathbb{R}^{r}$$ and $$p,q\in (0,\infty]$$ as \begin{align*} A_{\mathbf{s}}^{p,q}(D)&:=\mathrm{Hol}(D)\cap L_{\mathbf{s}}^{p,q}(D)\\ A_{\mathbf{s},0}^{p,q}(D)&:=\mathrm{Hol}(D)\cap L_{\mathbf{s},0}^{p,q}(D), \end{align*} where $$L_{\mathbf{s}}^{p,q}(D)$$ is the Hausdorff space associated with the space $\{f\colon D\rightarrow \mathbb{C}\colon f \text{\:measurable\:} \int_{\Omega}(\Delta_{\Omega}^{\mathbf{s}}\|f_{h}\|_{L^{p}(\mathcal{N}})^{q}d\nu_{\Omega}(h)<\infty\}$ and $$L_{\mathbf{s},0}^{p,q}(D)$$ is the closure of $$C_{c}(D)$$ in $$L_{\mathbf{s}}^{p,q}(D)$$. The measure $$\nu_{\Omega}$$ is $$\Delta_{\Omega}^{\mathbf{d}}\cdot \mathcal{H}^{m}$$ for an appropriate $$\mathbf{d}$$ and $$\mathcal{H}^{m}$$ the Hausdorff measure. The symbol $$\mathcal{N}$$ stands for $$E\times F$$ endowed with the group structure $(\zeta,x)(\zeta^{'},x^{'}):=(\zeta+\zeta^{'},x+x^{'}+2\Im \Phi(\zeta,\zeta^{'})$ and $f_{h}\colon \mathcal{N}\ni (\zeta,x)\mapsto f(\zeta,x+i\Phi(\zeta)+ih)\in \mathbb{C}.$ The values $$p,q,\mathbf{s}$$ for which $$A_{\mathbf{s}}^{p,q}(D)$$ is non-trivial are characterized (Proposition 3.5). Some sampling results are obtained (Theorems 3.22 and 3.23). In Section 3.4 the authors deal with atomic decomposition for the spaces $$A_{\mathbf{s}}^{p,q}(D)$$. In Section 3.5 there is studied duality of these spaces. In Chapter 4 the authors introduce and study Besov-type spaces $$B_{p,q}^{\mathbf{s}}(\mathcal{N},\Omega)$$. The theory parallels the classical one defined on $$\mathbb{R}^{n}$$. It is modelled in relation to the boundary values of the spaces $$A_{\mathbf{s}}^{p,q}(D)$$. It should be noted that the group $$\mathcal{N}$$ is not commutative and the authors deal with the full range of exponents $$p,q\in (0,\infty]$$. The main results of Chapter 5 concern the boundedness of the Bergman projectors. Theorem 5.25 gives some conditions on $$p,q\in [1,\infty]$$ and $$\mathbf{s}, \mathbf{s}^{'}\in \mathbb{R}^{r}$$ such that the corresponding projector $$P_{\mathbf{s}^{'}}$$ (defined in Definition 5.19) is a continuous linear mapping of $$L_{\mathbf{s},0}^{p,q}(D)$$ into $$\tilde{A}_{\mathbf{s}}^{p,q}(D)$$ (this space is defined as the image of some extension operator on appropriate Besov spaces -- see Definition 5.3). The study of the Bergman projectors is related to atomic decompositions. The chapter opens with a discussion of boundary values $$f_{h}$$ as $$h\rightarrow 0$$, $$h\in \Omega$$ for functions $$f\in A_{\mathbf{s}}^{p,q}(D)$$. Reviewer: Michal Jasiczak (Poznań) More about the Stieltjes function from which the discrete classical orthogonal polynomials are characterized https://www.zbmath.org/1485.33009 2022-06-24T15:10:38.853281Z "Lorente, Anier Soria" https://www.zbmath.org/authors/?q=ai:lorente.anier-soria "Centurión Fajardo, Alicia María" https://www.zbmath.org/authors/?q=ai:centurion-fajardo.alicia-maria "López Novoa, Fidel" https://www.zbmath.org/authors/?q=ai:lopez-novoa.fidel Summary: In this work we consider a new way of constructing the Stieltjes function from which the discrete classical orthogonal polynomials (Charlier, Krawtchouk, Meixner, and Hahn polynomials) are characterized. In addition, the hypergeometric series representations for the Stieltjes function are obtained for one discrete classical case. Sobolev orthogonal polynomials of several variables on product domains https://www.zbmath.org/1485.33010 2022-06-24T15:10:38.853281Z "Dueñas Ruiz, Herbert" https://www.zbmath.org/authors/?q=ai:duenas-ruiz.herbert "Salazar-Morales, Omar" https://www.zbmath.org/authors/?q=ai:salazar-morales.omar "Piñar, Miguel" https://www.zbmath.org/authors/?q=ai:pinar.miguel-a In [\textit{L. Fernández} et al., J. Comput. Appl. Math. 284, 202--215 (2015; Zbl 1312.33039)], polynomial bases orthogonal in terms of the weight $$w_{1} (x)w_{2}(y)$$ supported on the rectangle $$A= [a_{1}, b_{1}]\times [a_{2},b_{2}]$$ with respect to the inner product $\langle f, g\rangle = \int _{A} \nabla f(x, y) \cdot \nabla g(x, y)w_{1} (x)w_{2}(y) dxdy+\lambda f(c_{1}, c_{2})g(c_{1} ,c_{2}),$ where $$(c_{1},c_{2})$$ is a fixed point of the two-dimensional Euclidean space, are studied. The authors focus the attention on two particular examples, the product of two Laguerre and the product of two Gegenbauer weight functions, respectively with different choices of the parameters. The motivation for using this kind of inner product comes from problems of approximation of solutions of partial differential equations that require control over the gradient. In the contribution under review, the authors deal with a natural extension of the above paper by analyzing algebraic properties of multivariate orthogonal polynomials with respect to a discrete-continuous Sobolev inner product $\langle f, g\rangle _{S} = \sum_{j=0}^{\kappa-1} \lambda_{j} \nabla^{j}f(\textbf{p})\cdot \nabla^{j}g(\textbf{p})+ c\int_{\Omega} \nabla^{\kappa} f(\textbf{x})\cdot \nabla^{\kappa} g(\textbf{x}) W(\textbf{x}) d\textbf{x}.$ Here, $$\nabla^{j}f, j=0,1, \cdots, \kappa,$$ denotes the column vector containing all the partial derivatives of order $$j$$ of the function $$f$$, the weight function $$W(\textbf{x}) = w_{1}(x_{1}), \cdots w_{d}(x_{d})$$ is the product of $$d$$ weight functions $$w_{k}(x)$$ supported on intervals $$[a_{k}, b_{k}], k=1, 2, \cdots, k,$$ of the real line, $$\lambda_{j}> 0, j=0, \cdots, \kappa-1,$$ $$c$$ is the normalization constant of the weight $$W,$$ and $$\textbf{p}$$ is a point in the $$d$$-dimensional Euclidean space. In other words, they deal with a particular case of weight function supported on the parallelepiped $$\Omega= \{\textbf{x}=(x_{1}, x_{2}, \cdots, x_{d}), x_{k}\in (a_{k}, b_{k}) \}$$. A connection formula between an orthogonal polynomial basis with respect to the discrete-continuous Sobolev inner product $$\langle \cdot, \cdot \rangle _{S}$$ and an orthogonal polynomial basis with respect to the continuous component of the above inner product is deduced. Assuming each weight $$w_{k}, k=1, 2, \cdots, d,$$ is classical (Hermite, Laguerre, Jacobi) and following the ideas developed when $$d = 2$$ in [\textit{H. Dueñas Ruiz} et al., Integral Transforms Spec. Funct. 28, No. 12, 988--1008 (2017; Zbl 1379.33020)], a connection formula between the corresponding polynomial bases defined as column vectors is given. The matrix coefficients in such a connection formula are represented in a compact way. Finally, the above results are illustrated in the case $$d = 3$$ for the Hermite-Hermite-Laguerre product weight function. Reviewer: Francisco Marcellán (Leganes) Laguerre expansions on conic domains https://www.zbmath.org/1485.33011 2022-06-24T15:10:38.853281Z "Xu, Yuan" https://www.zbmath.org/authors/?q=ai:xu.yuan|xu.yuan.1 The author investigated the Fourier orthogonal expansions with respect to Laguerre type weight functions on the conic surface of revolution and the domain bounded by such a surface in this article. The main results are a closed form formula for the reproducing kernels of the orthogonal projection operator and a pseudo convolution structure on the conic domain, which is shown to be bounded in an appropriate $$L_{p}$$ space and is used to study the mean convergence of the Cesàro means of the Laguerre expansions on conic domains. Reviewer: D. L. Suthar (Dessie) Mittag-Leffler-Hyers-Ulam stability of differential equation using Fourier transform https://www.zbmath.org/1485.34180 2022-06-24T15:10:38.853281Z "Mohanapriya, Arusamy" https://www.zbmath.org/authors/?q=ai:mohanapriya.arusamy "Park, Choonkil" https://www.zbmath.org/authors/?q=ai:park.choonkil "Ganesh, Anumanthappa" https://www.zbmath.org/authors/?q=ai:ganesh.anumanthappa "Govindan, Vediyappan" https://www.zbmath.org/authors/?q=ai:govindan.vediyappan Summary: This research paper aims to present the results on the Mittag-Leffler-Hyers-Ulam and Mittag-Leffler-Hyers-Ulam-Rassias stability of linear differential equations of first, second, and $$n$$th order by the Fourier transform method. Moreover, the stability constant of such equations is obtained. Some examples are given to illustrate the main results. A mean value formula for the iterated Dunkl-Helmoltz operator https://www.zbmath.org/1485.35012 2022-06-24T15:10:38.853281Z "González Vieli, F. J." https://www.zbmath.org/authors/?q=ai:gonzalez-vieli.francisco-javier Summary: We establish a spherical mean value formula for the iterated Dunkl-Helmoltz operator, thus generalizing a result of Mejjaoli and Trimèche. We then give an application to distributions with Dunkl transform supported by the unit sphere. Strong traces to degenerate parabolic equations https://www.zbmath.org/1485.35274 2022-06-24T15:10:38.853281Z "Erceg, Marko" https://www.zbmath.org/authors/?q=ai:erceg.marko "Mitrović, Darko" https://www.zbmath.org/authors/?q=ai:mitrovic.darko Global Fourier integral operators in the plane and the square function https://www.zbmath.org/1485.35447 2022-06-24T15:10:38.853281Z "Manna, Ramesh" https://www.zbmath.org/authors/?q=ai:manna.ramesh "Ratnakumar, P. K." https://www.zbmath.org/authors/?q=ai:ratnakumar.p-k Summary: We prove the local smoothing estimate for general Fourier integral operators with phase function of the form $$\phi(x, t, \xi)=x\cdot\xi + t\,q(\xi)$$, with $$q \in C^\infty(\mathbb{R}^2\setminus\{0\})$$, homogeneous of degree one, and amplitude functions in the symbol class of order $$m \le 0$$. The result is global in the space variable, and also improves our previous work in this direction [the authors, in: Advances in harmonic analysis and partial differential equations. Based on the 12th ISAAC congress, session Harmonic analysis and partial differential equations'', Aveiro, Portugal, July 29 -- August 2, 2019. Cham: Birkhäuser. 1--35 (2020; Zbl 1477.42005)]. The approach involves a reduction to operators with amplitude function depending only on the covariable, and a new estimate for square function based on angular decomposition. Filled Julia sets of Chebyshev polynomials https://www.zbmath.org/1485.37043 2022-06-24T15:10:38.853281Z "Christiansen, Jacob Stordal" https://www.zbmath.org/authors/?q=ai:christiansen.jacob-stordal "Henriksen, Christian" https://www.zbmath.org/authors/?q=ai:henriksen.christian "Pedersen, Henrik Laurberg" https://www.zbmath.org/authors/?q=ai:pedersen.henrik-laurberg "Petersen, Carsten Lunde" https://www.zbmath.org/authors/?q=ai:petersen.carsten-lunde The authors study the limit behavior (taking into account the Hausdorff distance) of the filled Julia sets of dual Chebyshev polynomials related to a non-polar compact set $$K$$ of the complex plane. They prove versions of known results for the asymptotic theory of orthogonal polynomials, in particular in the case of Chebyshev polynomials. In particular, in the main theorem it is proved that the measures of maximal entropy for the sequence of dual Chebyshev polynomials of the compact $$K$$ converges weak* to the equilibrium measure on $$K$$. Reviewer: José Manuel Gutiérrez Jimenez (Logrono) Some basic properties and fundamental relations for discrete Muckenhoupt and Gehring classes https://www.zbmath.org/1485.40007 2022-06-24T15:10:38.853281Z "Saker, S. H." https://www.zbmath.org/authors/?q=ai:saker.saker-h|saker.samir-h "Rabie, S. S." https://www.zbmath.org/authors/?q=ai:rabie.safi-s "Alzabut, Jehad" https://www.zbmath.org/authors/?q=ai:alzabut.jehad-o "O'Regan, D." https://www.zbmath.org/authors/?q=ai:oregan.donal "Agarwal, R. P." https://www.zbmath.org/authors/?q=ai:agarwal.ravi-p Summary: In this paper, we prove some basic properties of the discrete Muckenhoupt class $$\mathcal{A}^p$$ and the discrete Gehring class $$\mathcal{G}^q$$. These properties involve the self-improving properties and the fundamental transitions and inclusions relations between the two classes. Approximation and localized polynomial frame on conic domains https://www.zbmath.org/1485.41006 2022-06-24T15:10:38.853281Z "Xu, Yuan" https://www.zbmath.org/authors/?q=ai:xu.yuan.1|xu.yuan Author's abstract: Highly localized kernels constructed by orthogonal polynomials have been fundamental in the recent development of approximation and computational analysis on the unit sphere, unit ball, and several other regular domains. In this work, we first study homogeneous spaces that are assumed to contain highly localized kernels and establish a framework for approximation and localized tight frames in such spaces, which extends recent works on bounded regular domains. We then show that the framework is applicable to homogeneous spaces defined on bounded conic domains, which consists of conic surfaces and the solid domains bounded by such surfaces and hyperplanes. The highly localized kernels on conic domains require precise estimates that rely on recently discovered addition formulas for orthogonal polynomials with respect to special weight functions on each domain and an intrinsic distance that takes into account the boundary of the domain, the latter is not comparable to the Euclidean distance at around the apex of the cone. The main results provide construction of semi-discrete localized tight frame in weighted $$L^2$$ norm and characterization of best approximation by polynomials on conic domains. The latter is achieved by using a $$K$$-functional, defined via the differential operator that has orthogonal polynomials as eigenfunctions, as well as a modulus of smoothness defined via a multiplier operator that is equivalent to the $$K$$-functional. Several intermediate results are of interest in their own right, including the Marcinkiewicz-Zygmund inequalities, positive cubature rules, Christoeffel functions, and Bernstein type inequalities. Moreover, although the highly localizable kernels hold only for special families of weight functions on each domain, many intermediate results are shown to hold for doubling weights defined via the intrinsic distance on the domain. Reviewer: Francisco Pérez Acosta (La Laguna) Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials https://www.zbmath.org/1485.41007 2022-06-24T15:10:38.853281Z "Zhang, Zhihua" https://www.zbmath.org/authors/?q=ai:zhang.zhihua|zhang.zhihua.1 Summary: Fourier approximation plays a key role in qualitative theory of deterministic and random differential equations. In this paper, we will develop a new approximation tool. For an $$m$$-order differentiable function $$f$$ on $$[0,1]$$, we will construct an $$m$$-degree algebraic polynomial $$P_m$$ depending on values of $$f$$ and its derivatives at ends of $$[0,1]$$ such that the Fourier coefficients of $$R_m =f-P_m$$ decay fast. Since the partial sum of Fourier series $$R_m$$ is a trigonometric polynomial, we can reconstruct the function $$f$$ well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes. On generalized characteristics of smoothness of functions and on average $$\nu$$-widths in the space $$L_2(\mathbb{R})$$ https://www.zbmath.org/1485.41014 2022-06-24T15:10:38.853281Z "Vakarchuk, S. B." https://www.zbmath.org/authors/?q=ai:vakarchuk.sergey-b "Vakarchuk, M. B." https://www.zbmath.org/authors/?q=ai:vakarchuk.mihail-b Summary: Estimates above and estimates below have been obtained for Kolmogorov, linear and Bernshtein average $$\nu$$-widths on the classes of functions $$W^r (\omega^w, \Psi)$$, where $$r \in \mathbb{N}$$, $$\omega^w(f)$$ is the generalized characteristic of smoothness of a function $$f \in L_2(\mathbb{R})$$, $$\Psi$$ is a majorant. Exact values of the enumerated extremal characteristics of approximation, following from the one condition on the majorant were obtained too. Approximation of function using generalized Zygmund class https://www.zbmath.org/1485.42001 2022-06-24T15:10:38.853281Z "Nigam, H. K." https://www.zbmath.org/authors/?q=ai:nigam.hare-krishna "Mursaleen, Mohammad" https://www.zbmath.org/authors/?q=ai:mursaleen.mohammad|mursaleen.mohammad-ayman "Rani, Supriya" https://www.zbmath.org/authors/?q=ai:rani.supriya Summary: In this paper we review some of the previous work done by \textit{M. V. Singh} et al. [J. Inequal. Appl. 2017, Paper No. 101, 11 p. (2017; Zbl 1360.42005)], \textit{S. Lal} and \textit{Shireen} [Bull. Math. Anal. Appl. 5, No. 4, 1--13 (2013; Zbl 1314.42012)], etc., on error approximation of a function $$g$$ in the generalized Zygmund space and resolve the issue of these works. We also determine the best error approximation of the functions $$g$$ and $$g^{\prime}$$, where $$g^{\prime}$$ is a derived function of a $$2 \pi$$-periodic function $$g$$, in the generalized Zygmund class $$X_z^{(\eta)}$$, $$z\geq 1$$, using matrix-Cesàro $$(TC^\delta)$$ means of its Fourier series and its derived Fourier series, respectively. Theorem 2.1 of the present paper generalizes eight earlier results, which become its particular cases. Thus, the results of \textit{B. P. Dhakal} [Int. Math. Forum 5, No. 33--36, 1729--1735 (2010; Zbl 1210.42002); Approximation of a function $$f$$ belonging to Lip class by $$(N, p, q)C_1$$ means of its Fourier series'', Int. J. Eng. Technol. 2, No. 3, 1--15 (2013)], the first author [Surv. Math. Appl. 5, 113--122 (2010; Zbl 1399.42005); Commun. Appl. Anal. 14, No. 4, 607--614 (2010; Zbl 1214.42012)], the first author and \textit{A. Sharma} [Kyungpook Math. J. 50, No. 4, 545--556 (2010; Zbl 1227.42008); Degree of approximation of a function belonging to $$\mathrm{Lip}(\xi,(t),r)$$ class by $$(E,1)(C,1)$$ product means'', Int. J. Pure Appl. Math. 70, No. 6, 775--784 (2011)], \textit{J. K. Kushwaha} and \textit{B. P. Dhakal} [Approximation of a function belonging to $$\mathrm{Lip} (\alpha, r)$$ class by $$N_{p,q}$$: $$C_1$$ summability method of its Fourier series'', Nepal J. Sci. Technol. 14, No. 2, 117--122 (2013; \url{doi:10.3126/njst.v14i2.10424})], \textit{U. K. Shrivastava} et al. [Approximation of function belonging to the $$\mathrm{Lip}(\Psi (t), p)$$ class by matrix-Cesáro summability method'', IOSR J. Math. 10, No. 1, 39--41 (2014)] become particular cases of our Theorem 2.1. Several corollaries are also deduced from our Theorem 2.1. Approximation of functions in generalized Zygmund class by double Hausdorff matrix https://www.zbmath.org/1485.42002 2022-06-24T15:10:38.853281Z "Nigam, H. K." https://www.zbmath.org/authors/?q=ai:nigam.hare-krishna "Mursaleen, M." https://www.zbmath.org/authors/?q=ai:mursaleen.mohammad-ayman|mursaleen.mohammad "Rani, Supriya" https://www.zbmath.org/authors/?q=ai:rani.supriya Summary: In the present work, we emphasize, for the first time, the error estimation of a two-variable function $$g(y,z)$$ in the generalized Zygmund class $$Y_r^{(\xi)}$$ ($$r\geq 1$$) using the double Hausdorff matrix means of its double Fourier series. In fact, in this work, we establish two theorems on error estimation of a two-variable function of $$g$$ in the generalized Zygmund class. Inverse theorems on the approximation of periodic functions with high generalized smoothness https://www.zbmath.org/1485.42003 2022-06-24T15:10:38.853281Z "Runovskii, K. V." https://www.zbmath.org/authors/?q=ai:runovski.konstantin-v "Laktionova, N. V." https://www.zbmath.org/authors/?q=ai:laktionova.n-v The authors proved the validity of the inverse estimate of Bernstein type $\left\Vert f^{\left( \lambda ,\beta \right) }\right\Vert _{p}\leq c\sum_{k=0}^{\infty }\left( \lambda _{k+1}-\lambda _{k}\right) E_{k}\left(f\right) _{p},\text{ }f\in L^{p}\text{ }\left( p\geq 1\right)$ and of its direct corollaries $E_{n-1}\left( f^{\left( \lambda ,\beta \right) }\right) _{p}\leq c\left(\lambda _{n}E_{n-1}\left( f\right) _{p}+\sum_{k=n}^{\infty }\left( \lambda _{k+1}-\lambda _{k}\right) E_{k}\left( f\right) _{p}\right)$ $\left\Vert T^{\left( \lambda ,\beta \right) }\right\Vert _{p}\leq c\lambda_{n}\left\Vert T\right\Vert _{p},\text{ }c\in\mathbb{C},\text{ }\lambda _{n}\nearrow \infty \text{, }\beta \in\mathbb{R},$ where $$E_{n}\left( f\right) _{p}$$ is the best approximation of a function $$f\in L^{p}$$ by trigonometric polynomials $$T$$ of degree at most $$n\in \mathbb{N}$$ and $\left( \cdot \right) ^{\left( \lambda ,\beta \right) }:e^{ikx}\rightarrow\lambda _{k}e^{sgnk\left( i\pi \beta \right) /2}e^{ikx},\text{ }k\in \mathbb{Z},$ is the linear operator of the $$\left( \lambda ,\beta \right)$$-derivative defined on the corresponding class of smooth functions in the generalized sense (as distributions). Reviewer: Włodzimierz Łenski (Poznań) Superoscillating sequences and supershifts for families of generalized functions https://www.zbmath.org/1485.42004 2022-06-24T15:10:38.853281Z "Colombo, F." https://www.zbmath.org/authors/?q=ai:colombo.fabrizio "Sabadini, I." https://www.zbmath.org/authors/?q=ai:sabadini.irene "Struppa, D. C." https://www.zbmath.org/authors/?q=ai:struppa.daniele-carlo "Yger, A." https://www.zbmath.org/authors/?q=ai:yger.alain Summary: We construct a large class of superoscillating sequences, more generally of $$\mathscr{F}$$-supershifts, where $$\mathscr{F}$$ is a family of smooth functions in $$(t, x)$$ (resp. distributions in $$(t, x)$$, or hyperfunctions in $$x$$ depending on the parameter $$t$$) indexed by $$\lambda \in\mathbb{R}$$. The frame in which we introduce such families is that of the evolution through Schrödinger equation $$(i\partial/\partial t - \mathscr{H}(x))(\psi )=0$$ ($$\mathscr{H}(x) = -(\partial^2/\partial x^2)/2 + V(x))$$, $$V$$ being a suitable potential). If $$\mathscr{F}= \{(t, x) \mapsto \varphi_\lambda (t, x); \lambda \in\mathbb{R}\}$$, where $$\varphi_\lambda$$ is evolved from the initial datum $$x\mapsto e^{i\lambda x}$$, $$\mathscr{F}$$-supershifts will be of the form $$\{\sum_{j=0}^N C_j(N, a) \varphi_{1-2j/N}\}_{N\ge 1}$$ for $$a\in\mathbb{R}\setminus[-1, 1]$$, taking $$C_j(N, a) =\binom{N}{j}(1+a)^{N-j}(1-a)^j/2^N$$. Our results rely on the fact that integral operators of the Fresnel type govern, as in optical diffraction, the evolution through the Schrödinger equation, such operators acting continuously on the weighted algebra of entire functions $$\mathrm{Exp}(\mathbb{C})$$. Analyzing in particular the quantum harmonic oscillator case forces us, in order to take into account singularities of the evolved datum that occur when the stationary phasis in the Fresnel operator vanishes, to enlarge the notion of $$\mathscr{F}$$-supershift, $$\mathscr{F}$$ being a family of $$C^\infty$$ functions or distributions in $$(t, x)$$, to that where $$\mathscr{F}$$ is a family of hyperfunctions in $$x$$, depending on $$t$$ as a parameter. Fourier expansions for higher-order Apostol-Genocchi, Apostol-Bernoulli and Apostol-Euler polynomials https://www.zbmath.org/1485.42005 2022-06-24T15:10:38.853281Z "Corcino, Cristina B." https://www.zbmath.org/authors/?q=ai:corcino.cristina-bordaje "Corcino, Roberto B." https://www.zbmath.org/authors/?q=ai:corcino.roberto-bagsarsa Summary: Fourier expansions of higher-order Apostol-Genocchi and Apostol-Bernoulli polynomials are obtained using Laurent series and residues. The Fourier expansion of higher-order Apostol-Euler polynomials is obtained as a consequence. Trigonometric series and self-similar sets https://www.zbmath.org/1485.42006 2022-06-24T15:10:38.853281Z "Li, Jialun" https://www.zbmath.org/authors/?q=ai:li.jialun "Sahlsten, Tuomas" https://www.zbmath.org/authors/?q=ai:sahlsten.tuomas Summary: Let $${F}$$ be a self-similar set on $$\mathbb{R}$$ associated to contractions $$f_j(x)=r_jx+b_j$$, $$j\in\mathcal{A}$$, for some finite $$\mathcal{A}$$, such that $$F$$ is not a singleton. We prove that if $$\log r_i / \log r_j$$ is irrational for some $$i\neq j$$, then $$F$$ is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of $$F$$. No separation conditions are assumed on $$F$$. We establish our result by showing that every self-similar measure $$\mu$$ on $$F$$ is a Rajchman measure: the Fourier transform $$\widehat{\mu}(\xi)\to 0$$ as $$|\xi|\to\infty$$. The rate of $$\widehat{\mu}(\xi)\to 0$$ is also shown to be logarithmic if $$\log r_i / \log r_j$$ is diophantine for some $$i\neq j$$. The proof is based on quantitative renewal theorems for stopping times of random walks on $$\mathbb{R}$$. The Fourier transform of a function related to cylinder functions and asymptotic expansions with logarithmic terms https://www.zbmath.org/1485.42007 2022-06-24T15:10:38.853281Z "Gorenflo, Norbert" https://www.zbmath.org/authors/?q=ai:gorenflo.norbert In this paper, the author considers the function $$E_0$$, given by $E_0:=H_0(x)-\frac{2i}{\pi}{J}_0(x)\ln\left(\frac{x}{2}\right),$ where $$H_0$$ is the first kind Hankel function of order zero and $${J}_0$$ is the Bessel function of order zero. It is found that since the logarithmic singularity of the Hankel function $$H_0$$ is removed, the function $$E_0$$ can be continued to an entire function. The author uses the function $$E_0$$ to derive asymptotic expansions of certain other functions, especially of some solutions of differential equations, whose asymptotics contain logarithmic terms too. Further, the author obtains the Fourier transform of the function $$E_0(k\cdot)$$, given by $E_0(k\cdot)^{\hat{}}(\rho)=\left[\frac{4i}{\pi}(\ln(k^2-\rho^2)-2\ln(k)+2\ln(2)+\gamma)+2\right]\frac{\chi_{(-k,k)}(\rho)}{\sqrt{k^2-\rho^2}},\,\rho\in\mathbb{R}$by examining a fifth-order ordinary differential equation which can be viewed as a generalization of the differential equation for the zero-order cylinder functions. Reviewer: Shiv K. Kaushik (Delhi) Block-cyclic structuring of the basis of Fourier transforms based on cyclic substitution https://www.zbmath.org/1485.42008 2022-06-24T15:10:38.853281Z "Prots'ko, I." https://www.zbmath.org/authors/?q=ai:protsko.i "Mishchuk, M." https://www.zbmath.org/authors/?q=ai:mishchuk.m Summary: The use of substitution as a primitive element in forming a cyclic basis matrix of the Fourier transform is considered. A cyclic substitution is used for block-cyclic structuring of the harmonic basis, which allows synthesizing the algorithms for fast discrete Fourier transforms of arbitrary size based on cyclic convolutions. The rearrangement of the cycles order and their first elements in cyclic substitutions is shown to reduce the amount of computation of cyclic convolutions in fast algorithms for the discrete Fourier transforms. On the stability of Fourier phase retrieval https://www.zbmath.org/1485.42009 2022-06-24T15:10:38.853281Z "Steinerberger, Stefan" https://www.zbmath.org/authors/?q=ai:steinerberger.stefan Summary: Phase retrieval is concerned with recovering a function $$f$$ from the absolute value of its Fourier transform $$|\widehat{f}|$$. We study the stability properties of this problem in Lebesgue spaces. Our main results shows that $\|f-g\|_{L^2(\mathbb{R}^n)} \le 2\cdot \| |\widehat{f}| - |\widehat{g}| \|_{L^2(\mathbb{R}^n)} + h_f\Bigl(\|f-g\|_{L^p(\mathbb{R}^n)}\Bigr) + J(\widehat{f}, \widehat{g}),$ where $$1 \le p < 2$$, $$h_f$$ is an explicit nonlinear function depending on the smoothness of $$f$$ and $$J$$ is an explicit term capturing the invariance under translations. A noteworthy aspect is that the stability is phrased in terms of $$L^p$$ for $$1 \le p < 2$$: in this region $$L^p$$ cannot be used to control $$L^2$$, our stability estimate has the flavor of an inverse Hölder inequality. It seems conceivable that the estimate is optimal up to constants. Direct and inverse theorems on the approximation of almost periodic functions in Besicovitch-Stepanets spaces https://www.zbmath.org/1485.42010 2022-06-24T15:10:38.853281Z "Serdyuk, A. S." https://www.zbmath.org/authors/?q=ai:serdyuk.anatoly-s "Shidlich, A. L." https://www.zbmath.org/authors/?q=ai:shidlich.andrii The authors deduce direct and inverse theorems on the approximation of functions from the Besicovitch-Stepanets spaces (i.e., from all Besicovitch almost periodic functions of order 1 for which the sums of the $$p$$th degrees of absolute values of their Fourier coefficients are finite), in terms of their best approximations and generalized moduli of smoothness. Reviewer: Luis Filipe Pinheiro de Castro (Aveiro) Approximation properties of the double Fourier sphere method https://www.zbmath.org/1485.42011 2022-06-24T15:10:38.853281Z "Mildenberger, Sophie" https://www.zbmath.org/authors/?q=ai:mildenberger.sophie "Quellmalz, Michael" https://www.zbmath.org/authors/?q=ai:quellmalz.michael Summary: We investigate analytic properties of the double Fourier sphere (DFS) method, which transforms a function defined on the two-dimensional sphere to a function defined on the two-dimensional torus. Then the resulting function can be written as a Fourier series yielding an approximation of the original function. We show that the DFS method preserves smoothness: it continuously maps spherical Hölder spaces into the respective spaces on the torus, but it does not preserve spherical Sobolev spaces in the same manner. Furthermore, we prove sufficient conditions for the absolute convergence of the resulting series expansion on the sphere as well as results on the speed of convergence. Trigonometric approximation of functions $$f(x,y)$$ of generalized Lipschitz class by double Hausdorff matrix summability method https://www.zbmath.org/1485.42012 2022-06-24T15:10:38.853281Z "Mishra, Abhishek" https://www.zbmath.org/authors/?q=ai:mishra.abhishek-c "Mishra, Vishnu Narayan" https://www.zbmath.org/authors/?q=ai:mishra.vishnu-narayan "Mursaleen, M." https://www.zbmath.org/authors/?q=ai:mursaleen.mohammad Summary: In this paper, we establish a new estimate for the degree of approximation of functions $$f(x,y)$$ belonging to the generalized Lipschitz class $$\mathrm{Lip} ((\xi_1, \xi_2);r)$$, $$r \geq 1$$, by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from $$\mathrm{Lip} ((\alpha,\beta);r)$$ and $$\mathrm{Lip}(\alpha,\beta)$$ in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and $$(C, \gamma, \delta)$$ means. Quantum Fourier analysis https://www.zbmath.org/1485.42013 2022-06-24T15:10:38.853281Z "Jaffe, Arthur" https://www.zbmath.org/authors/?q=ai:jaffe.arthur-m "Jiang, Chunlan" https://www.zbmath.org/authors/?q=ai:jiang.chunlan "Wu, Jinsong" https://www.zbmath.org/authors/?q=ai:wu.jinsong Summary: Quantum Fourier analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform, as a map between suitably defined $$L^p$$ spaces, leading to an uncertainty principle for relative entropy. We cite several applications of quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a topological inequality, and we outline several open problems. On the regularity of distributions via the convergence of the continuous shearlet transform in two dimensions https://www.zbmath.org/1485.42014 2022-06-24T15:10:38.853281Z "Navarro, Jaime" https://www.zbmath.org/authors/?q=ai:navarro.jaime "Elizarraraz, David" https://www.zbmath.org/authors/?q=ai:elizarraraz.david Fourier restriction above rectangles https://www.zbmath.org/1485.42015 2022-06-24T15:10:38.853281Z "Schwend, Jeremy" https://www.zbmath.org/authors/?q=ai:schwend.jeremy "Stovall, Betsy" https://www.zbmath.org/authors/?q=ai:stovall.betsy Summary: In this article, we study the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We state a conjecture for the dependence of the operator norms in these inequalities on the sidelengths of the rectangles, prove that this conjecture follows from (a slight reformulation of the) restriction conjecture for elliptic hypersurfaces, and prove that, if valid, the conjecture is essentially sharp. Such questions arise naturally in the study of restriction inequalities for degenerate hypersurfaces; we demonstrate this connection by using our positive results to prove new restriction inequalities for a class of hypersurfaces having some additive structure. The Schrödinger equation in $$L^p$$ spaces for operators with heat kernel satisfying Poisson type bounds https://www.zbmath.org/1485.42016 2022-06-24T15:10:38.853281Z "Chen, Peng" https://www.zbmath.org/authors/?q=ai:chen.peng "Duong, Xuan Thinh" https://www.zbmath.org/authors/?q=ai:duong.xuan-thinh "Fan, Zhijie" https://www.zbmath.org/authors/?q=ai:fan.zhijie "Li, Ji" https://www.zbmath.org/authors/?q=ai:li.ji.1 "Yan, Lixin" https://www.zbmath.org/authors/?q=ai:yan.lixin The operator $$e^{it\Delta}$$ is $$L^p(\mathbb{R}^n) \mapsto L^p(\mathbb{R}^n)$$ bounded only when $$p = 2$$, but the $$p$$ range can be widened prior regularization. This type of phenomenon was already investigated, for example, in [\textit{T. A. Bui} et al., Rev. Mat. Iberoam. 36, No. 2, 455--484 (2020; Zbl 1448.35352)]. In the article under review, the authors consider mapping properties of $$e^{itL}$$ for non-negative self-adjoint operators $$L$$ in a metric space $$(X, d, \mu)$$ with doubling measure $$\mu$$. Their main result is the inequality $\lVert e^{itL}(I + L)^{-\sigma_pn}f \rVert_{L^p(X)} \le C(I + \lvert t \rvert)^{\sigma_pn}\lVert f \rVert_{L^p(X)}, \qquad \text{for } \sigma_p := \bigg\lvert \frac{1}{2} - \frac{1}{p} \bigg\rvert, \tag{1}$ where $$p \in (p_0, p_0^\prime)$$ with $$1\le p_0 < 2$$, and $$e^{-tL}$$ satisfies $\lVert \1_{B(x,t^{1/m})}e^{-tL}V_{t^{1/m}}^{\sigma_{p_0}} \1_{B(y,t^{1/m})} f \rVert_{L^2(X)} \le C \Big(1 + \frac{d(x,y)}{t^{1/m}}\Big)^{-n - \kappa} \lVert f \rVert_{L^{p_0}(X)}, \tag{2}$ where $$\kappa > [n/2] + 1$$ and $$V_r(x) := \mu(B(x,r))$$. The hypothesis (2) covers a wide class of operators. The authors develop a theory of Hardy spaces $$H_L^q(X)$$ adapted to $$L$$ and show that, by duality and interpolation, it suffices to prove (1) with $$H_L^q(X)$$, for $$q < 1$$, instead of $$L^p(X)$$. To show that the operator is bounded in $$H^q_L(X)$$, they prove new estimates for oscillatory multipliers to control off-diagonal terms. The proof uses several techniques, for example: $$L^2$$ based estimates; a dyadic-like decomposition (amalgam blocks); and commutator estimates. \par Editorial note: The reviewer found some mistakes in the original paper [see \url{arXiv:2007.01469}] and contacted the authors. They uploaded an amended version of the paper in [\url{arXiv:2007.01469}]. The review above relates to the amended version. Reviewer: Felipe Ponce-Vanegas (Bilbao) Multiplier conditions for boundedness into Hardy spaces https://www.zbmath.org/1485.42017 2022-06-24T15:10:38.853281Z "Grafakos, Loukas" https://www.zbmath.org/authors/?q=ai:grafakos.loukas "Nakamura, Shohei" https://www.zbmath.org/authors/?q=ai:nakamura.shohei "Nguyen, Hanh Van" https://www.zbmath.org/authors/?q=ai:van-nguyen.hanh "Sawano, Yoshihiro" https://www.zbmath.org/authors/?q=ai:sawano.yoshihiro Summary: In the present work we find useful and explicit necessary and sufficient conditions for linear and multilinear multiplier operators of Coifman-Meyer type, finite sum of products of Calderón-Zygmund operators, and also of intermediate types to be bounded from a product of Lebesgue or Hardy spaces into a Hardy space. These conditions state that the symbols of the multipliers $$\sigma (\xi_1,\dots,\xi_m)$$ and their derivatives vanish on the hyperplane $$\xi_1+\cdots +\xi_m=0$$. The multilinear Hörmander multiplier theorem with a Lorentz-Sobolev condition https://www.zbmath.org/1485.42018 2022-06-24T15:10:38.853281Z "Grafakos, Loukas" https://www.zbmath.org/authors/?q=ai:grafakos.loukas "Park, Bae Jun" https://www.zbmath.org/authors/?q=ai:park.bae-jun Summary: In this article, we provide a multilinear version of the Hörmander multiplier theorem with a Lorentz-Sobolev space condition. The work is motivated by the recent result of \textit{L. Grafakos} and \textit{L. Slavíková} [Int. Math. Res. Not. 2019, No. 15, 4764--4783 (2019; Zbl 1459.42013)] where an analogous version of classical Hörmander multiplier theorem was obtained; this version is sharp in many ways and reduces the number of indices that appear in the statement of the theorem. As a natural extension of the linear case, in this work, we prove that if $$mn/2<s<mn$$, then \begin{aligned} \Vert T_{\sigma }(f_1,\dots ,f_m) \Vert_{L^p(\mathbb{R}^n)}\lesssim \sup_{k\in \mathbb{Z}} \Vert \sigma (2^k\ \cdot)\widehat{\Psi^{(m)}} \Vert_{L_s^{mn/s,1}(\mathbb{R}^{mn})}\Vert f_1\Vert_{L^{p_1}(\mathbb{R}^n)}\cdots \Vert f_m\Vert_{L^{p_m}(\mathbb{R}^n)} \end{aligned} for certain $$p,p_1,\dots ,p_m$$ with $$1/p=1/p_1+\dots +1/p_m$$. We also show that the above estimate is sharp, in the sense that the Lorentz-Sobolev space $$L_s^{mn/s,1}$$ cannot be replaced by $$L_s^{r,q}$$ for $$r<mn/s, 0<q\le \infty$$, or by $$L_s^{mn/s,q}$$ for $$q>1$$. Riesz means on locally symmetric spaces https://www.zbmath.org/1485.42019 2022-06-24T15:10:38.853281Z "Papageorgiou, Effie" https://www.zbmath.org/authors/?q=ai:papageorgiou.effie-g Summary: We prove that for a certain class of $$n$$ dimensional rank one locally symmetric spaces, if $$f \in L^p$$, $$1\le p \le 2$$, then the Riesz means of order $$z$$ of $$f$$ converge to $$f$$ almost everywhere, for $$\Re z > (n-1)(1/p-1/2)$$. Fourier multipliers on a vector-valued function space https://www.zbmath.org/1485.42020 2022-06-24T15:10:38.853281Z "Park, Bae Jun" https://www.zbmath.org/authors/?q=ai:park.bae-jun Summary: We study multiplier theorems on a vector-valued function space, which is a generalization of the results of \textit{A. P. Calderon} and \textit{A. Torchinsky} [Adv. Math. 24, 101--171 (1977; Zbl 0355.46021)], and \textit{L. Grafakos} et al. [Ill. J. Math. 61, No. 1--2, 25--35 (2017; Zbl 1395.42025)], and an improvement of the result of \textit{H. Triebel} [J. Approx. Theory 28, 317--328 (1980; Zbl 0446.46018)]. For $$0<p<\infty$$ and $$0<q\le \infty$$ we obtain that if $$r>\frac{d}{s-(d/\min{(1,p,q)}-d)}$$, then $\big \Vert \big \{\big(m_k \widehat{f_k}\big)^{\vee}\big\}_{k\in{\mathbb{Z}}}\big \Vert_{L^p(\ell^q)}\lesssim_{p,q} \sup_{l\in{\mathbb{Z}}}{\big \Vert m_l(2^l\cdot)\big \Vert_{L_s^r({\mathbb{R}}^d)}} \big \Vert \big \{f_k\big\}_{k\in{\mathbb{Z}}}\big \Vert_{L^p(\ell^q)}, \quad f_k\in{\mathcal{E}}(A2^k),$ under the condition $$\max{(|d/p-d/2|,|d/q-d/2|)}<s<d/\min{(1,p,q)}$$. An extension to $$p=\infty$$ will be additionally considered in the scale of Triebel-Lizorkin space. Our result is sharp in the sense that the Sobolev space in the above estimate cannot be replaced by Sobolev spaces $$L_s^r$$ with $$r\le \frac{d}{s-(d/\min{(1,p,q)}-d)}$$. Bochner-Riesz operators between Morrey spaces https://www.zbmath.org/1485.42021 2022-06-24T15:10:38.853281Z "Wang, Hua" https://www.zbmath.org/authors/?q=ai:wang.hua.1|wang.hua|wang.hua.2 "Xiao, Jie" https://www.zbmath.org/authors/?q=ai:xiao.jie|xiao.jie.1|xiao.jie.2 "Xu, Shaozhen" https://www.zbmath.org/authors/?q=ai:xu.shaozhen Summary: This note concerns the boundedness of $$\mathscr{J}_\delta$$ (the Bochner-Riesz operator) mapping $$L^{p,\kappa}$$ (the $$(p,\kappa)$$-Morrey space) to $$L^{q,\lambda}$$ (the $$(q,\lambda)$$-Morrey space) or $$L^{q,\lambda;\ln}$$ (the $$(q,\lambda;\ln)$$-Morrey space), thereby showing $\|\mathscr{J}_\delta f\|_{L^{p,\lambda}}\lesssim\|f\|_{L^{p,\kappa}}\;\forall\;f\in L^{p,\kappa}\;\text{ under }\; \begin{cases} n\ge\kappa>\lambda>0;&\\ 1\le p<\infty;&\\ \delta\ge\frac{n-1}2+\frac{\lambda-\kappa}p, \end{cases}$ which may be regarded as the Morrey ($$\kappa>\lambda)$$-variant of the unsolved Bochner-Riesz conjecture (cf. [\textit{C. Benea} et al., Trans. Lond. Math. Soc. 4, No. 1, 110--128 (2017; Zbl 1395.42021)] or [\textit{E. M. Stein}, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy. Princeton, NJ: Princeton University Press (1993; Zbl 0821.42001), p. 390]): $\|\mathscr{J}_\delta f\|_{L^p}\lesssim\|f\|_{L^p}\;\forall\; f\in L^p\;\text{ under }\; 2\ne p\in(1,\infty)\;\;\&\;\;\delta>n\left|\frac1p-\frac12\right|-\frac12.$ Quantitative weighted bounds for Calderón commutators with rough kernels https://www.zbmath.org/1485.42022 2022-06-24T15:10:38.853281Z "Chen, Yanping" https://www.zbmath.org/authors/?q=ai:chen.yanping.2|chen.yanping.3|chen.yanping.1 "Li, Ji" https://www.zbmath.org/authors/?q=ai:li.ji.1|li.ji.3|li.ji.2|li.ji.4 Summary: We obtain a quantitative weighted bound for the Calderón commutator $$\mathcal{C}_\Omega$$ which is a typical example of a non-convolution Calderón-Zygmund operator under the condition $$\Omega \in L^\infty (\mathbb{S}^{n-1})$$; this is the best known quantitative result for this class of rough operators. Upper and lower bounds for Littlewood-Paley square functions in the Dunkl Setting https://www.zbmath.org/1485.42023 2022-06-24T15:10:38.853281Z "Dziubański, Jacek" https://www.zbmath.org/authors/?q=ai:dziubanski.jacek "Hejna, Agnieszka" https://www.zbmath.org/authors/?q=ai:hejna.agnieszka In this paper the authors prove some integral bounds for Littlewood-Paley square functions in the Dunkl context. In $$\mathbb R^N$$ endowed with a normalized root system $$R$$ and a multiplicity function $$k\geq 0$$, the authors consider the classical gradient $$\nabla$$, the Dunkl gradient $$\nabla_k$$, the Dunkl Laplacian $$\Delta_k$$ and the carré du champ operator $$\Gamma$$ associated to $$\Delta_k$$. By means of two fixed functions $$\Phi$$ and $$\Psi$$ (not necessarily radial), defined on $$\mathbb R^N$$ and satisfying certain smoothness, integrability and decay conditions, the authors introduce some square functions, which are associated in a natural way to $$\nabla_k$$, $$\Delta_k$$, $$\Gamma$$. By adapting some tecniques from Calderón-Zygmund analysis to the Dunkl framework, they first prove an upper bound for the sum of the $$L^p$$-norms (with respect to a measure, suitably defined in terms of the root system and the multiplicity function) of the square functions associated to $$\nabla$$, $$\nabla_k$$, $$\Gamma$$. Lower bounds for the Dunkl square functions are also proved under an additional condition, stating essentially that the Dunkl transforms of $$\Phi$$ and $$\Psi$$ are not identically zero along any direction. Reviewer: Valentina Casarino (Vicenza) Hardy type estimates for Riesz transforms associated with Schrödinger operators on the Heisenberg group https://www.zbmath.org/1485.42024 2022-06-24T15:10:38.853281Z "Gao, Chunfang" https://www.zbmath.org/authors/?q=ai:gao.chunfang Summary: Let $$\mathbb{H}^n$$ be the Heisenberg group and $$Q=2n+2$$ be its homogeneous dimension. Let $$\mathcal{L}=-\Delta_{\mathbb{H}^n}+V$$ be the Schrödinger operator on $$\mathbb{H}^n$$, where $$\Delta_{\mathbb{H}^n}$$ is the sub-Laplacian and the nonnegative potential $$V$$ belongs to the reverse Hölder class $$B_{q_1}$$ for $$q_1\geq Q/2$$. Let $${H_{\mathcal{L}}^p(\mathbb{H}^n)}$$ be the Hardy space associated with the Schrödinger operator $$\mathcal{L}$$ for $$Q/(Q+\delta_0) < p \leq 1$$, where $$\delta_0 = \min\{1,2-Q/q_1\}$$. In this paper we consider the Hardy type estimates for the operator $$T_\alpha =V^\alpha (-\Delta_{\mathbb{H}^n} + V)^{-\alpha}$$, and the commutator $$[b,T_\alpha]$$, where $$0 < \alpha < Q/2$$. We prove that $$T_\alpha$$ is bounded from $$H_{\mathcal{L}}^p (\mathbb{H}^n)$$ into $$L^p(\mathbb{H}^n)$$. Suppose that $$b \in BMO_{\mathcal{L}}^\theta (\mathbb{H}^n)$$, which is larger than $$BMO(\mathbb{H}^n)$$. We show that the commutator $$[b,T_\alpha]$$ is bounded from $$H_\mathcal{L}^1 (\mathbb{H}^n)$$ into weak $$L^1(\mathbb{H}^n)$$. Boundedness and compactness of commutators associated with Lipschitz functions https://www.zbmath.org/1485.42025 2022-06-24T15:10:38.853281Z "Guo, Weichao" https://www.zbmath.org/authors/?q=ai:guo.weichao "He, Jianxun" https://www.zbmath.org/authors/?q=ai:he.jianxun|he.jianxun.1 "Wu, Huoxiong" https://www.zbmath.org/authors/?q=ai:wu.huoxiong "Yang, Dongyong" https://www.zbmath.org/authors/?q=ai:yang.dongyong Given $$\alpha\in[0,1]$$, $$f\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$$ and a cube $$Q$$, let $$f_Q=\frac1{|Q|}\int_Q f$$; $$Q$$ always stands for a cube with sides parallel to the coordinate axes. The space $$\mathrm{BMO}_\alpha (\mathbb{R}^n)$$ of functions with fractional mean oscillation consists of all $$f\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$$ such that $\|f\|_{\mathrm{BMO}_\alpha (\mathbb{R}^n)}:=\sup_Q \frac1{|Q|^{1+\frac {\alpha}{n}}}\int_Q|f-f_Q|<\infty.$ For $$\alpha=0$$ this is the usual BMO space on $$\mathbb{R}^n$$. For $$\beta\in[0,n)$$ and $$\Omega$$, a real-valued homogeneous function of degree 0 on $$\mathbb{R}^n$$, consider the operator defined for suitable functions $$f$$ by $T_{\Omega ,\beta} f(x)=\int_{\mathbb R^n}\frac{\Omega(x-y)}{|x-y|^{n-\beta}}f(y)\,dy;$ for $$\beta=0$$, when $$\int_{S^{n-1}}\Omega \,d\sigma=0$$ is additionally assumed, $$T_{\Omega ,\beta}$$ is a singular integral operator; if $$\beta\in(0,n)$$ and $$\Omega \equiv1$$, then $$T_{\Omega ,\beta}$$ is a fractional integral operator. For $$b\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$$ consider the commutator $$T^1_bf:=[b,T_{\Omega,\beta}]f=bT_{\Omega,\beta} f-T_{\Omega,\beta}(bf)$$ and the iterated commmutators $$T^m_b=[b,T^{m-1}_b]f$$, $$m\ge2$$ (in this case it is adittionally asumed that the symbol $$b$$ is real-valued). In the paper the authors study the $$(L^p,L^q)$$-compactness of the commutators $$T^m_b$$. An essential ingredient in this study consists in using $$\mathrm{CMO}$$ type spaces $$\mathrm{CMO}_\alpha(\mathbb{R}^n)$$. For a fixed $$\alpha\in[0,1]$$, $$\mathrm{CMO}_\alpha(\mathbb{R}^n)$$ is the closure of $$C^\infty_c(\mathbb R^n)$$ in $$\mathrm{BMO}_\alpha(\mathbb{R}^n)$$, and a useful characterization of functions from this space in terms of the mean value oscillation is obtained by the authors when $$\alpha\in[0,1)$$. In fact this extends an earlier result of \textit{A. Uchiyama} [Tohoku Math. J. (2) 30, 163--171 (1978; Zbl 0384.47023)] corresponding to $$\alpha=0$$, i.e. the classical BMO case. In one of the two main theorems of the paper it is proved that, under suitable assumptions on $$p$$, $$q$$, $$\alpha$$, $$\beta$$, $$m$$, $$\Omega$$ and a weight function $$\omega$$, the commutator $$T^m_b$$ is a compact operator from $$L^p(\omega^p)$$ to $$L^q(\omega^q)$$ if and only if $$b\in \mathrm{CMO}_\alpha(\mathbb{R}^n)$$. Reviewer: Krzysztof Stempak (Wrocław) Operator-free sparse domination https://www.zbmath.org/1485.42026 2022-06-24T15:10:38.853281Z "Lerner, Andrei K." https://www.zbmath.org/authors/?q=ai:lerner.andrei-k "Lorist, Emiel" https://www.zbmath.org/authors/?q=ai:lorist.emiel "Ombrosi, Sheldy" https://www.zbmath.org/authors/?q=ai:ombrosi.sheldy-j Summary: We obtain a sparse domination principle for an arbitrary family of functions $$f(x,Q)$$, where $$x\in\mathbb{R}^n$$ and $$Q$$ is a cube in $$\mathbb{R}^n$$. When applied to operators, this result recovers our recent works [\textit{A. K. Lerner} and \textit{S. Ombrosi}, J. Geom. Anal. 30, No. 1, 1011--1027 (2020; Zbl 1434.42020); \textit{E. Lorist}, J. Geom. Anal. 31, No. 9, 9366--9405 (2021; Zbl 1477.42016)]. On the other hand, our sparse domination principle can be also applied to non-operator objects. In particular, we show applications to generalised Poincaré-Sobolev inequalities, tent spaces and general dyadic sums. Moreover, the flexibility of our result allows us to treat operators that are not localisable in the sense of [Lorist, loc. cit.], as we will demonstrate in an application to vector-valued square functions. $$L^p$$ boundedness of Carleson \& Hilbert transforms along plane curves with certain curvature constraints https://www.zbmath.org/1485.42027 2022-06-24T15:10:38.853281Z "Li, Junfeng" https://www.zbmath.org/authors/?q=ai:li.junfeng "Yu, Haixia" https://www.zbmath.org/authors/?q=ai:yu.haixia In the main result of this paper, which is Theorem 1.1, the authors show that if $$p\in(1,\infty)$$, $$u:\mathbb{R}\rightarrow\mathbb{R}$$ is a measurable function and $$\gamma$$ is a plane curve with certain constraints then the Carleson transform $\mathcal{C}_{u,\gamma}f(x)=p.v.\int_{-\infty}^{\infty}e^{iu(x)\gamma(t)}f(x-t)\frac{dt}{t}\qquad x\in\mathbb{R}$ and the Hilbert transform $H_{u,\gamma}f(x_{1},x_{2})=p.v.\int_{-\infty}^{\infty}f(x_{1}-t,x_{2}-u(x_{1})\gamma(t))\frac{dt}{t}\qquad(x_{1},x_{2})\in\mathbb{R}^{2}$ are bounded on $$L^{p}$$. Reviewer: Israel Pablo Rivera Ríos (Bahía Blanca) Rough singular integrals and maximal operator with radial-angular integrability https://www.zbmath.org/1485.42028 2022-06-24T15:10:38.853281Z "Liu, Ronghui" https://www.zbmath.org/authors/?q=ai:liu.ronghui "Wu, Huoxiong" https://www.zbmath.org/authors/?q=ai:wu.huoxiong This paper refers to the boundedness of the Calderón-Zygmund operator $T_\Omega f(x) = \text{p.v.} \int_{\mathbb{R}^n} f(x-y) \frac{\Omega(y')}{|y|^n} \, dy,$ where $$y^\prime = y/|y|$$ and the kernel $$\Omega$$ satisfies $$\int_{S^{n-1}} \Omega(y) d \sigma(y) = 0$$, $$S^{n-1}$$ being the unit sphere in $$\mathbb{R}^n$$, for functions $$f$$ in the mixed radial-angular space $$L^p_{|x|} L^{\tilde{p}}_\theta (\mathbb{R}^n)$$, for any $$1\leq p, \, \tilde{p} \leq \infty$$. For these kind of funcions, the boundedness of $$T_\Omega$$ was previously obtained for $$\Omega \in \mathcal{C}^1 (S^{n-1})$$ and also for $$\Omega \in L \log L (S^{n-1})$$. Here one considers the case that $$\Omega \in H^1(S^{n-1})$$, the Hardy space on $$S^{n-1}$$. We recall that the space $$L^p_{|x|} L^{\tilde{p}}_\theta (\mathbb{R}^n)$$ is formed by those funcions $$f$$ for which $\|f\|_{L^p_{|x|} L^{\tilde{p}}_\theta (\mathbb{R}^n)} := \bigg( \int_0^\infty \|f(r \cdot)\|^p_{L^{\tilde{p}} (S^{n-1})} \, r^{n-1} \, dr \bigg)^{1/p} < \infty .$ In this paper one proves that for $$\Omega \in H^1(S^{n-1})$$ with $$\int_{S^{n-1}} \Omega(y) d \sigma(y) = 0$$ and $$1 < \tilde{p} \leq p < \tilde{p} n / (n-1)$$ or $$\tilde{p} n /(\tilde{p} +n-1) <p\leq \tilde{p} < \infty$$, the following inequality holds, $\|T_\Omega f\|_{L^p_{|x|} L^{\tilde{p}}_\theta (\mathbb{R}^n)} \leq C_{p, \tilde{p}} \|\Omega\|_{H^1 (S^{n-1})} \|f\|_{L^p_{|x|} L^{\tilde{p}}_\theta (\mathbb{R}^n)}.$ An analogous inequality is stablished for the maximal singular integral operator, $$T^\ast_\Omega$$, given by $T^\ast_\Omega f(x) = \sup_{\varepsilon >0} \bigg| \int_{|y|\geq \varepsilon} f(x-y) \frac{\Omega(y^\prime)}{|y|^n} \, dy\bigg|,$ when $$1 < \tilde{p} \leq p < \tilde{p} n /(n-1)$$. Reviewer: Julià Cufí (Bellaterra) A nonlinear version of Roth's theorem on sets of fractional dimensions https://www.zbmath.org/1485.42029 2022-06-24T15:10:38.853281Z "Li, Xiang" https://www.zbmath.org/authors/?q=ai:li.xiang.2|li.xiang.3|li.xiang.4 "He, Qianjun" https://www.zbmath.org/authors/?q=ai:he.qianjun "Yan, Dunyan" https://www.zbmath.org/authors/?q=ai:yan.dunyan "Zhang, Xingsong" https://www.zbmath.org/authors/?q=ai:zhang.xingsong Summary: Let $$E\subset \mathbb{R}$$ be a closed set, which has nonzero Hausdorff dimension and some other properties. For some $$t>0$$, we proved the three- point patterns $$x$$, $$x+t$$, $$x+\gamma (t)$$ belong to $$E$$, where $$\gamma (t)$$ is a convex curve with some curvature constraints. Removable singularities for Lipschitz caloric functions in time varying domains https://www.zbmath.org/1485.42030 2022-06-24T15:10:38.853281Z "Mateu, Joan" https://www.zbmath.org/authors/?q=ai:mateu.joan "Prat, Laura" https://www.zbmath.org/authors/?q=ai:prat.laura "Tolsa, Xavier" https://www.zbmath.org/authors/?q=ai:tolsa.xavier Summary: In this paper we study removable singularities for regular $$(1,1/2)$$-Lipschitz solutions of the heat equation in time varying domains. We introduce an associated Lipschitz caloric capacity and we study its metric and geometric properties and the connection with the $$L^2$$ boundedness of the singular integral whose kernel is given by the gradient of the fundamental solution of the heat equation. Small cap decoupling inequalities: bilinear methods https://www.zbmath.org/1485.42031 2022-06-24T15:10:38.853281Z "Oh, Changkeun" https://www.zbmath.org/authors/?q=ai:oh.changkeun Summary: We obtain sharp small cap decoupling inequalities associated to the moment curve for certain range of exponents $$p$$. Our method is based on the bi-linearization argument due to \textit{J. Bourgain} and \textit{C. Demeter} [Ann. Math. (2) 182, No. 1, 351--389 (2015; Zbl 1322.42014)]. Our result generalizes theirs to all higher dimensions. Boundedness for commutators of rough $$p$$-adic Hardy operator on $$p$$-adic central Morrey spaces https://www.zbmath.org/1485.42032 2022-06-24T15:10:38.853281Z "Sarfraz, Naqash" https://www.zbmath.org/authors/?q=ai:sarfraz.naqash "Aslam, Muhammad" https://www.zbmath.org/authors/?q=ai:aslam.muhammad-saeed|aslam.muhammad-jamil|aslam.muhammad-kamran|aslam.muhammad-zubair|aslam.muhammad-nauman|aslam.muhammad-shamrooz "Jarad, Fahd" https://www.zbmath.org/authors/?q=ai:jarad.fahd Summary: In the present article we obtain the boundedness for commutators of rough $$p$$-adic Hardy operator on $$p$$-adic central Morrey spaces. Furthermore, we also acquire the boundedness of rough $$p$$-adic Hardy operator on Lebesgue spaces. Energy counterexamples in two weight Calderón-Zygmund theory https://www.zbmath.org/1485.42033 2022-06-24T15:10:38.853281Z "Sawyer, Eric T." https://www.zbmath.org/authors/?q=ai:sawyer.eric-t "Shen, Chun-Yen" https://www.zbmath.org/authors/?q=ai:shen.chun-yen "Uriarte-Tuero, Ignacio" https://www.zbmath.org/authors/?q=ai:uriarte-tuero.ignacio Summary: We show that the energy conditions are not necessary for boundedness of Riesz transforms in dimension $$n\geq 2$$. In dimension $$n=1$$, we construct an elliptic singular integral operator $$H_{\flat}$$ for which the energy conditions are not necessary for boundedness of $$H_{\flat}$$. The convolution kernel $$K_{\flat}(x)$$ of the operator $$H_{\flat}$$ is a smooth flattened version of the Hilbert transform kernel $$K(x)=\frac{1}{x}$$ that satisfies ellipticity $$\vert K_{\flat}(x)\vert\gtrsim\frac{1}{\vert x\vert}$$, but not gradient ellipticity $$\vert K^\prime_{\flat}(x)\vert\gtrsim\frac{1}{\vert x\vert^2}$$. Indeed the kernel has flat spots where $$K^\prime_{\flat}(x)=0$$ on a family of intervals, but $$K^\prime_{\flat}(x)$$ is otherwise negative on $$\mathbb{R}\setminus\{0\}$$. On the other hand, if a one-dimensional kernel $$K(x,y)$$ is both elliptic and gradient elliptic, then the energy conditions are necessary, and so by our theorem in [\textit{E. T. Sawyer} et al., in: Harmonic analysis, partial differential equations and applications. In honor of Richard L. Wheeden. Basel: Birkhäuser/Springer. 125--164 (2017; Zbl 1380.42015)], the $$T1$$ theorem holds for such kernels on the line. This paper includes results from [\textit{E. T. Sawyer}, Energy conditions and twisted localizations of operators'', Preprint, \url{arXiv:1801.03706v2}]. Sharp weak type estimates for a family of Soria bases https://www.zbmath.org/1485.42034 2022-06-24T15:10:38.853281Z "Dmitrishin, Dmitry" https://www.zbmath.org/authors/?q=ai:dmitrishin.dmitry "Hagelstein, Paul" https://www.zbmath.org/authors/?q=ai:hagelstein.paul-alton "Stokolos, Alex" https://www.zbmath.org/authors/?q=ai:stokolos.alex Summary: Let $${\mathcal{B}}$$ be a collection of rectangular parallelepipeds in $${\mathbb{R}}^3$$ whose sides are parallel to the coordinate axes and such that $${\mathcal{B}}$$ contains parallelepipeds with side lengths of the form $$s$$, $$\frac{2^N}{s}$$, $$t$$, where $$s, t > 0$$ and $$N$$ lies in a nonempty subset $$S$$ of the natural numbers. We show that if $$S$$ is an infinite set, then the associated geometric maximal operator $$M_{\mathcal{B}}$$ satisfies the weak type estimate $\left| \left\{ x \in{\mathbb{R}}^3 : M_{{\mathcal{B}}}f(x) > \alpha \right\} \right| \le C \int \nolimits_{{\mathbb{R}}^3} \frac{|f|}{\alpha} \left( 1 + \log^+ \frac{|f|}{\alpha}\right)^2,$ but does not satisfy an estimate of the form $\left| \left\{ x \in{\mathbb{R}}^3 : M_{{\mathcal{B}}}f(x) > \alpha \right\} \right| \le C \int \nolimits_{{\mathbb{R}}^3} \phi \left( \frac{|f|}{\alpha}\right)$ for any convex increasing function $$\phi : \mathbb [0, \infty) \rightarrow [0, \infty)$$ satisfying the condition $\lim_{x \rightarrow \infty}\frac{\phi (x)}{x (\log (1 + x))^2} = 0\;.$ Correction to: A weak reverse Hölder inequality for caloric measure'' https://www.zbmath.org/1485.42035 2022-06-24T15:10:38.853281Z "Genschaw, Alyssa" https://www.zbmath.org/authors/?q=ai:genschaw.alyssa "Hofmann, Steve" https://www.zbmath.org/authors/?q=ai:hofmann.steve From the text: In our paper [ibid. 30, No. 2, 1530--1564 (2020; Zbl 1436.42028)], in which we presented a parabolic version of results of [\textit{B. Bennewitz} and \textit{J. L. Lewis}, Complex Variables, Theory Appl. 49, No. 7--9, 571--582 (2004; Zbl 1068.31001)], the proof of Lemma 2.7 (the main lemma) was divided into several cases, one of which we had inadvertently failed to note and treat. We now rectify this omission. Fortunately, this will be a simple matter. Extrapolation and the boundedness in grand variable exponent Lebesgue spaces without assuming the log-Hölder continuity condition, and applications https://www.zbmath.org/1485.42036 2022-06-24T15:10:38.853281Z "Kokilashvili, Vakhtang" https://www.zbmath.org/authors/?q=ai:kokilashvili.vakhtang-m "Meskhi, Alexander" https://www.zbmath.org/authors/?q=ai:meskhi.alexander Summary: The boundedness of the Hardy-Littlewood maximal operator, and the weighted extrapolation in grand variable exponent Lebesgue spaces are established provided that Hardy-Littlewood maximal operator is bounded in appropriate variable exponent Lebesgue space. Moreover, we give some bounds of the norm of the Hardy-Littlewood maximal operator in these spaces. As corollaries, we have appropriate norm inequalities and the boundedness of operators of Harmonic Analysis such as maximal and sharp maximal functions; Calderón-Zygmund singular integrals, commutators of singular integrals in grand variable exponent Lebesgue spaces. Finally, applying the boundedness results of integral operators of Harmonic Analysis, we have the direct and inverse theorems on the approximation of $$2\pi$$-periodic functions by trigonometric polynomials in the framework of grand variable exponent Lebesgue spaces. Convergence problem of Schrödinger equation in Fourier-Lebesgue spaces with rough data and random data https://www.zbmath.org/1485.42037 2022-06-24T15:10:38.853281Z "Yan, Xiangqian" https://www.zbmath.org/authors/?q=ai:yan.xiangqian "Zhao, Yajuan" https://www.zbmath.org/authors/?q=ai:zhao.yajuan "Yan, Wei" https://www.zbmath.org/authors/?q=ai:yan.wei Summary: In this paper, we consider the convergence problem of Schrödinger equation. Firstly, we show the almost everywhere pointwise convergence of Schrödinger equation in Fourier-Lebesgue spaces $$\hat{H}^{\frac{1}{p},\frac{p}{2}}(\mathbb{R})$$ $$(4\leq p<\infty)$$, $$\hat{H}^{\frac{3s_1}{p},\frac{2p}{3}}(\mathbb{R}^2)$$ $$(s_1>\frac{1}{3},\ 3\leq p<\infty)$$, $$\hat{H}^{\frac{2s_2}{p},p}(\mathbb{R}^n)$$ $$(s_2>\frac{n}{2(n+1)},\ 2\leq p<\infty,\ n\geq 3)$$ with rough data. Secondly, we show that the maximal function estimate related to one dimensional Schrödinger equation can fail with data in $$\hat{H}^{s,\frac{p}{2}}(\mathbb{R})$$ $$(s<\frac{1}{p})$$. Finally, we show the stochastic continuity of Schrödinger equation with random data in $$\hat{L}^r(\mathbb{R}^n)$$ $$(2\leq r<\infty)$$ almost surely. The main ingredients are maximal function estimates and density theorem in Fourier-Lebesgue spaces as well as some large deviation estimates. Endpoint regularity of discrete multilinear fractional nontangential maximal functions https://www.zbmath.org/1485.42038 2022-06-24T15:10:38.853281Z "Zhang, Daiqing" https://www.zbmath.org/authors/?q=ai:zhang.daiqing Summary: Given $$m\geq 1$$, $$0\leq \lambda \leq 1$$, and a discrete vector-valued function $$\vec{f}=(f_1,\dots,f_m)$$ with each $$f_j:\mathbb{Z}^d\rightarrow \mathbb{R}$$, we consider the discrete multilinear fractional nontangential maximal operator $\mathrm{M}_{\alpha,\mathcal{B}}^{\lambda}(\vec{f}) (\vec{n})=\mathop{\sup_{r>0, \vec{x}\in \mathbb{R}^d}}_{\vert \vec{n}-\vec{x} \vert \leq \lambda r}\frac{1}{N(B_r(\vec{x}))^{m-\frac{\alpha}{d}}} \prod_{j=1}^m\sum_{\vec{k}\in B_r(\vec{x})\cap \mathbb{Z}^d} \bigl\vert f_j(\vec{k}) \bigr\vert,$ where $$\mathcal{B}$$ is the collection of all open balls $$B\subset \mathbb{R}^d$$, $$B_r(\vec{x})$$ is the open ball in $$\mathbb{R}^d$$ centered at $$\vec{x}\in \mathbb{R}^d$$ with radius $$r$$, and $$N(B_r(\vec{x}))$$ is the number of lattice points in the set $$B_r(\vec{x})$$. We show that the operator $$\vec{f}\mapsto |\nabla \mathrm{M}_{\alpha, \mathcal{B}}^{\lambda}(\vec{f})|$$ is bounded and continuous from $$\ell^1(\mathbb{Z}^d)\times \ell^1(\mathbb{Z}^d)\times \cdots \times \ell^1(\mathbb{Z}^d)$$ to $$\ell^q(\mathbb{Z}^d)$$ if $$0\leq \alpha < md$$ and $$q\geq 1$$ such that $$q>\frac{d}{md- \alpha +1}$$. We also prove that the same result also holds for the discrete multilinear fractional nontangential maximal operators associated with cubes. These results we obtained represent significant and natural extensions of what was known previously. Riesz transform characterizations for multidimensional Hardy spaces https://www.zbmath.org/1485.42039 2022-06-24T15:10:38.853281Z "Kania-Strojec, Edyta" https://www.zbmath.org/authors/?q=ai:kania-strojec.edyta "Preisner, Marcin" https://www.zbmath.org/authors/?q=ai:preisner.marcin Summary: We study Hardy space $$H^1_L(X)$$ related to a self-adjoint operator $$L$$ defined on an Euclidean subspace $$X$$ of $${{\mathbb{R}}^d}$$. We continue study from [\textit{E. Kania-Strojec} et al., Rev. Mat. Complut. 34, No. 2, 409--434 (2021; Zbl 1481.42029)], where, under certain assumptions on the heat semigroup $$\exp (-tL)$$, the atomic characterization of local type for $$H^1_L(X)$$ was proved. In this paper we provide additional assumptions that lead to another characterization of $$H^1_L(X)$$ by the Riesz transforms related to $$L$$. As an application, we prove the Riesz transform characterization of $$H^1_L(X)$$ for multidimensional Bessel and Laguerre operators, and the Dirichlet Laplacian on $${\mathbb{R}}^d_+$$. BMO type space associated with Neumann operator and application to a class of parabolic equations https://www.zbmath.org/1485.42040 2022-06-24T15:10:38.853281Z "Chao, Zhang" https://www.zbmath.org/authors/?q=ai:chao.zhang "Yang, Minghua" https://www.zbmath.org/authors/?q=ai:yang.minghua Let $$\mathrm{BMO}_{\Delta_{N}}(\mathbb{R}^n)$$ denote a BMO space on $$\mathbb{R}^n$$ associated to a Neumann operator. In this paper the authors show that a function $$f \in \mathrm{BMO}_{\Delta_{N}}(\mathbb{R}^n)$$ is the trace of $$\mathcal{L}u= u_{t}-\Delta_{N}u=0$$, $$u(x,0)=f(x)$$ where $$u$$ satisfies a Carleson-type condition $\sup_{x_{B}, r_{B}}r_{B}^{-n}\int_{0}^{r_{B}^2}\int_{B(x_{B}, r_{B})}\{ t|\partial_{t} u(x,t)|^2+|\nabla_{x}u(x,t)|^2\}dx dt < \infty.$ Conversely, this Carleson condition characterizes all the $$\mathcal{L}$$-carolic functions whose traces belong to the space $$\mathrm{BMO}_{\Delta_{N}}(\mathbb{R}^n)$$. Furthermore, by this characterization the authors prove the global well-posedness for parabolic equations of Navier-Stokes type with the Neumann boundary condition under smallness condition on the initial data in $$\mathrm{BMO}^{-1}_{\Delta_{N}}(\mathbb{R}^n)$$. Reviewer: Koichi Saka (Akita) Correction to: A sharp Bernstein-type inequality and application to the Carleson embedding theorem with matrix weights'' https://www.zbmath.org/1485.42041 2022-06-24T15:10:38.853281Z "Kraus, Daniela" https://www.zbmath.org/authors/?q=ai:kraus.daniela "Moucha, Annika" https://www.zbmath.org/authors/?q=ai:moucha.annika "Roth, Oliver" https://www.zbmath.org/authors/?q=ai:roth.oliver From the text: The original published version of this article [the authors, ibid. 12, No. 1, Paper No. 40, 6 p. (2022; Zbl 1482.42063)] contained a number of typos that were introduced in the typesetting process by the publisher. This update corrects those errors. Median-type John-Nirenberg space in metric measure spaces https://www.zbmath.org/1485.42042 2022-06-24T15:10:38.853281Z "Myyryläinen, Kim" https://www.zbmath.org/authors/?q=ai:myyrylainen.kim Summary: We study the so-called John-Nirenberg space that is a generalization of functions of bounded mean oscillation in the setting of metric measure spaces with a doubling measure. Our main results are local and global John-Nirenberg inequalities, which give weak-type estimates for the oscillation of a function. We consider medians instead of integral averages throughout, and thus functions are not a priori assumed to be locally integrable. Our arguments are based on a Calderón-Zygmund decomposition and a good-$$\lambda$$ inequality for medians. A John-Nirenberg inequality up to the boundary is proven by using chaining arguments. As a consequence, the integral-type and the median-type John-Nirenberg spaces coincide under a Boman-type chaining assumption. Invertibility of frame operators on Besov-type decomposition spaces https://www.zbmath.org/1485.42043 2022-06-24T15:10:38.853281Z "Romero, José Luis" https://www.zbmath.org/authors/?q=ai:romero.jose-luis "van Velthoven, Jordy Timo" https://www.zbmath.org/authors/?q=ai:van-velthoven.jordy-timo "Voigtlaender, Felix" https://www.zbmath.org/authors/?q=ai:voigtlaender.felix Summary: We derive an extension of the Walnut-Daubechies criterion for the invertibility of frame operators. The criterion concerns general reproducing systems and Besov-type spaces. As an application, we conclude that $$L^2$$ frame expansions associated with smooth and fast-decaying reproducing systems on sufficiently fine lattices extend to Besov-type spaces. This simplifies and improves recent results on the existence of atomic decompositions, which only provide a particular dual reproducing system with suitable properties. In contrast, we conclude that the $$L^2$$ canonical frame expansions extend to many other function spaces, and, therefore, operations such as analyzing using the frame, thresholding the resulting coefficients, and then synthesizing using the canonical dual frame are bounded on these spaces. The CMO-Dirichlet problem for the Schrödinger equation in the upper half-space and characterizations of CMO https://www.zbmath.org/1485.42044 2022-06-24T15:10:38.853281Z "Song, Liang" https://www.zbmath.org/authors/?q=ai:song.liang "Wu, Liangchuan" https://www.zbmath.org/authors/?q=ai:wu.liangchuan Summary: Let $$\mathcal{L}$$ be a Schrödinger operator of the form $$\mathcal{L}=-\Delta +V$$ acting on $$L^2(\mathbb{R}^n)$$ where the non-negative potential $$V$$ belongs to the reverse Hölder class $$\mathrm{RH}_q$$ for some $$q\ge (n+1)/2$$. Let $$\mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n)$$ denote the function space of vanishing mean oscillation associated to $$\mathcal{L}$$. In this article, we will show that a function $$f$$ of $$\mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n)$$ is the trace of the solution to $$\mathbb{L}u=-u_{tt}+\mathcal{L}u=0, u(x,0)=f(x)$$, if and only if, $$u$$ satisfies a Carleson condition $\sup_{B: \text{balls}}\mathcal{C}_{u,B} :=\sup_{B(x_B,r_B): \text{balls}} r_B^{-n}\int_0^{r_B}\int_{B(x_B, r_B)} \big |t \nabla u(x,t)\big |^2\, \frac{\mathrm{dx}\, \mathrm{dt}}{t} <\infty,$ and $\lim_{a \rightarrow 0}\sup_{B: r_B \le a} \,\mathcal{C}_{u,B} = \lim_{a \rightarrow \infty}\sup_{B: r_B \ge a} \,\mathcal{C}_{u,B} = \lim_{a \rightarrow \infty}\sup_{B: B \subseteq \left( B(0, a)\right)^c} \,\mathcal{C}_{u,B}=0.$ This continues the lines of the previous characterizations by \textit{X. T. Duong} et al. [J. Funct. Anal. 266, No. 4, 2053--2085 (2014; Zbl 1292.35099)] and \textit{R. Jiang} and \textit{B. Li} [On the Dirichlet problem for the Schrödinger equation with boundary value in BMO space'', Preprint, \url{arXiv:2006.05248}] for the $$\mathrm{BMO}_{\mathcal{L}}$$ spaces, which were founded by \textit{E. B. Fabes} et al. [Indiana Univ. Math. J. 25, 159--170 (1976; Zbl 0306.46032)] for the classical BMO space. For this purpose, we will prove two new characterizations of the $$\mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n)$$ space, in terms of mean oscillation and the theory of tent spaces, respectively. Asymptotics of Muttalib-Borodin determinants with Fisher-Hartwig singularities https://www.zbmath.org/1485.42045 2022-06-24T15:10:38.853281Z "Charlier, Christophe" https://www.zbmath.org/authors/?q=ai:charlier.christophe Summary: Muttalib-Borodin determinants are generalizations of Hankel determinants and depend on a parameter $$\theta >0$$. In this paper, we obtain large $$n$$ asymptotics for $$n \times n$$ Muttalib-Borodin determinants whose weight possesses an arbitrary number of Fisher-Hartwig singularities. As a corollary, we obtain asymptotics for the expectation and variance of the real and imaginary parts of the logarithm of the underlying characteristic polynomial, several central limit theorems, and some global bulk rigidity upper bounds. Our results are valid for all $$\theta > 0$$. Construction and implementation of asymptotic expansions for Laguerre-type orthogonal polynomials https://www.zbmath.org/1485.42046 2022-06-24T15:10:38.853281Z "Huybrechs, Daan" https://www.zbmath.org/authors/?q=ai:huybrechs.daan "Opsomer, Peter" https://www.zbmath.org/authors/?q=ai:opsomer.peter Summary: Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval $$[0,\infty)$$ with respect to a weight function of the form \begin{aligned} w(x) = x^{\alpha} e^{-Q(x)},\quad Q(x) =\sum_{k=0}^m q_k x^k, \quad \alpha > -1, \quad q_m > 0. \end{aligned} The classical Laguerre polynomials correspond to $$Q(x)=x$$. The computation of higher order terms of the asymptotic expansions of these polynomials for large degree becomes quite complicated, and a full description seems to be lacking in literature. However, this information is implicitly available in the work of \textit{M. Vanlessen} [Constr. Approx. 25, No. 2, 125--175 (2007; Zbl 1117.15025)], based on a nonlinear steepest descent analysis of an associated Riemann-Hilbert problem. We will extend this work and show how to efficiently compute an arbitrary number of higher order terms in the asymptotic expansions of Laguerre and Laguerre-type polynomials. This effort is similar to the case of Jacobi and Jacobi-type polynomials in a previous article. We supply an implementation with explicit expansions in four different regions of the complex plane. These expansions can also be extended to Hermite-type weights of the form $$\exp(-\sum_{k=0}^m q_k x^{2k})$$ on $$(-\infty,\infty)$$ and to general nonpolynomial functions $$Q(x)$$ using contour integrals. The expansions may be used, e.g., to compute Gauss-Laguerre quadrature rules with lower computational complexity than based on the recurrence relation and with improved accuracy for large degree. They are also of interest in random matrix theory. Parameters of a positive chain sequence associated with orthogonal polynomials https://www.zbmath.org/1485.42047 2022-06-24T15:10:38.853281Z "Marcato, Gustavo A." https://www.zbmath.org/authors/?q=ai:marcato.gustavo-a "Ranga, A. Sri" https://www.zbmath.org/authors/?q=ai:ranga.a-sri "Lun, Yen Chi" https://www.zbmath.org/authors/?q=ai:lun.yen-chi Summary: The objective here is to provide a new characterization of all the parameter sequences of a positive chain sequence that has been of importance in the study of orthogonal polynomials on the real line. Connection formulas and some applications that follow from this new characterization have also been explored. Example of a class of positive chain sequences having explicit formulas for all the parameter sequences is also provided. $$G$$-frames, fusion frames and the restricted isometry property https://www.zbmath.org/1485.42048 2022-06-24T15:10:38.853281Z "Asgari, Mohammad Sadegh" https://www.zbmath.org/authors/?q=ai:asgari.mohammad-sadegh "Kavian, Golsa" https://www.zbmath.org/authors/?q=ai:kavian.golsa Summary: In this paper we study the restricted isometry property for $$g$$-frames, we will show how to use tight $$g$$-frames that have the restricted isometry property to construct fusion frames. We also study the conditions which under removing some element from a $$g$$-frame, again we obtain another $$g$$-frame so that Theorem 4.3 obtained in [\textit{M. S. Asgari}, Acta Math. Sci., Ser. B, Engl. Ed. 31, No. 4, 1633--1642 (2011; Zbl 1249.42016)] is a special case of it. Operator parameterizations of frame generators and generalized dual pair of frame generators of unitary systems https://www.zbmath.org/1485.42049 2022-06-24T15:10:38.853281Z "Guo, Xunxiang" https://www.zbmath.org/authors/?q=ai:guo.xunxiang Let $$H$$ be a Hilbert space, and let $$B(H)$$ be the set of all bounded linear operators on $$H.$$ A set of unitary operators $$\mathcal{U}$$ acting on $$H$$ and containing the identity operator of $$B(H)$$ is called a unitary system. A vector $$x \in H$$ with the property that $$\mathcal{U}x := \{Ux: U \in \mathcal{U} \}$$ is a Bessel sequence for $$H$$ is called a Bessel generator of $$\mathcal{U}.$$ If $$x$$ is unit norm and $$\mathcal{U} x$$ is an orthonormal basis, then $$x$$ is called a complete wandering vector for $$\mathcal{U}.$$ If $$\mathcal{U} x$$ forms a frame for $$H,$$ we call $$x$$ a frame generator for $$\mathcal{U}.$$ Normalized tight frame generators and Riesz generators for $$\mathcal{U}$$ can be defined similarly. Other authors have studied complete wandering vectors and frame generators for unitary systems. Operator parametrizations of all complete wandering vectors and frame generators of a unitary system was also found earlier. Similar results on multi-wandering vectors and multi-frame vectors for unitary systems have also been obtained. However, these results rely on the assumption that there exist complete wandering vectors for unitary systems. However, this is not always the case. In this paper, the author first gives a condition under which a countable unitary system has complete wandering vectors. This is followed by general operator parametrizations of frame generators, normalized tight frame generators, and Riesz generators. Lastly, they introduce the concept of generalized dual frame generator pairs for a unitary system. Some properties and constructions of generalized dual frame generator pairs are also studied. Reviewer: Somantika Datta (Moscow) Characterization and wavelet packets associated with VN-MRA on $$L^2(K,\mathbb{C}^N)$$ https://www.zbmath.org/1485.42050 2022-06-24T15:10:38.853281Z "Bhat, M. Younus" https://www.zbmath.org/authors/?q=ai:bhat.mohammad-younus The author considers wavelet packets associated with nonuniform vector-valued wavelets on local fields of positive characteristic (see [\textit{M. Y. Bhat}, Complex Anal. Oper. Theory 13, No. 5, 2203--2228 (2019; Zbl 1418.42050)]). Reviewer: Yuri A. Farkov (Moskva) Multivariate quasi-tight framelets with high balancing orders derived from any compactly supported refinable vector functions https://www.zbmath.org/1485.42051 2022-06-24T15:10:38.853281Z "Han, Bin" https://www.zbmath.org/authors/?q=ai:han.bin.1|han.bin "Lu, Ran" https://www.zbmath.org/authors/?q=ai:lu.ran This paper is devoted to a construction of multivariate framelets (wavelet frames) which satisfy several desired properties. Most known framelets are constructed from refinable vector functions through the oblique extension principle (OEP) [\textit{C. K. Chui} et al., Appl. Comput. Harmon. Anal. 13, No. 3, 224--262 (2002; Zbl 1016.42023); \textit{I. Daubechies} et al., ibid. 14, No. 1, 1--46 (2003; Zbl 1035.42031); \textit{B. Han}, Framelets and wavelets. Algorithms, analysis, and applications. Cham: Birkhäuser (2017; Zbl 1387.42001)]. Starting with a dilation matrix $$M$$ and specific matrix-valued filter $$a$$, and a compactly supported $$M-$$refinable vector valued function $$\phi$$, the authors construct an OEP-based compactly supported quasi-tight framelet systems with specific properties. This includes, for example, the highest possible order of vanishing moments of framelet generators, and the highest balancing order of the associated fast framelet transform. The last property being essential for the sparsity of the framelet transform. One of the main results is a newly developed normal form of a multivariate matrix-valued filter, which greatly facilitates the study of refinable vector functions and related framelets. This improves some previous results of the first author given in [J. Approx. Theory 124, No. 1, 44--88 (2003; Zbl 1028.42019); Math. Comput. 79, No. 270, 917--951 (2010; Zbl 1247.42040)], and plays a key role in the study of multivariate quasi-tight multiframelets with high vanishing moments and the balancing property. In addition, the authors prove a multidimensional version of one of their previous results from [\textit{B. Han} and \textit{R. Lu}, Appl. Comput. Harmon. Anal. 51, 295--332 (2021; Zbl 1460.42055)]. Reviewer: Nenad Teofanov (Novi Sad) Uncertainty principles in term of supports in Hankel wavelet setting https://www.zbmath.org/1485.42052 2022-06-24T15:10:38.853281Z "Hkimi, S." https://www.zbmath.org/authors/?q=ai:hkimi.siwar "Omri, S." https://www.zbmath.org/authors/?q=ai:omri.slim The authors present firstly some properties of the Hankel transform and of the continuous Hankel wavelet transform. The main results are two uncertainty principles for the continuous Hankel wavelet transform. The first uncertainty principle states that the continuous Hankel wavelet transform of a function $$f \not= 0$$ can not be supported in a set of finite measure. The second uncertainty principle characterizes the annihilating set for the continuous Hankel transform. Reviewer: Manfred Tasche (Rostock) Continuous wavelet transforms for vector-valued functions https://www.zbmath.org/1485.42053 2022-06-24T15:10:38.853281Z "Ishi, Hideyuki" https://www.zbmath.org/authors/?q=ai:ishi.hideyuki "Oshiro, Kazuhide" https://www.zbmath.org/authors/?q=ai:oshiro.kazuhide Summary: We consider continuous wavelet transforms associated to unitary representations of the semi-direct product of a vector group with a linear Lie group realized on the Hilbert spaces of square-integrable vector-valued functions. In particular, we give a concrete example of an admissible vector-valued function (vector field) for the 3-dimensional similitude group. For the entire collection see [Zbl 1482.94007]. Mexican hat wavelet transform and its applications https://www.zbmath.org/1485.42054 2022-06-24T15:10:38.853281Z "Singh, Abhishek" https://www.zbmath.org/authors/?q=ai:singh.abhishek-kumar|singh.abhishek|singh.abhishek-kr "Rawat, Aparna" https://www.zbmath.org/authors/?q=ai:rawat.aparna "Raghuthaman, Nikhila" https://www.zbmath.org/authors/?q=ai:raghuthaman.nikhila This paper considers the Mexican hat wavelet which is one of the basic wavelet functions defined by the second derivative of a Gaussian function. The mexican hat wavelet is considered the most suitable mother wavelet function in many research papers. In this paper the Mexican hat wavelet theory (MHWT) is used to define the Mexican hat wavelet Stieltjes transform (MHWST) of a bounded variation function which can analyze both continuous and discrete-time signals. Further, the authors develop some necessary and sufficient conditions for representing functions as MHWT and MHWST. The introduction of a new representation of the wavelet transform and its relation with the Hilbert transform, fractional integral operators, and the Watson transform are used to derive certain roundedness results, approximation results, and some general Parseval formula for the wavelet transform. The authors provide a sufficient number of recent references. For the entire collection see [Zbl 1471.93009]. Reviewer: Haydar Akca (Abu Dhabi) Weak nonhomogeneous wavelet dual frames for Walsh reducing subspace of $$L^2(\mathbb{R}_+)$$ https://www.zbmath.org/1485.42055 2022-06-24T15:10:38.853281Z "Zhang, Yan" https://www.zbmath.org/authors/?q=ai:zhang.yan.5|zhang.yan.4 "Li, Yun-Zhang" https://www.zbmath.org/authors/?q=ai:li.yunzhang.1|li.yunzhang Generalized spherical mean value operators on Euclidean space https://www.zbmath.org/1485.45003 2022-06-24T15:10:38.853281Z "Okada, Yasunori" https://www.zbmath.org/authors/?q=ai:okada.yasunori "Yamane, Hideshi" https://www.zbmath.org/authors/?q=ai:yamane.hideshi The authors show that certain convolution equations can be seen as an extension of linear partial differential equations with certain coefficients. In the introduction, several citations confirm that such equations have various applications, as for example in the theory of holomorphic functions on convex domains. \textit{K.-L. Lim} [The spherical mean value operators on Euclidean and hyperbolic spaces. Medford, MA: Tufts University (PhD thesis) (2012)] proved that spherical mean value operators on Euclidean spaces are surjective convolution operators. In Section 2, the authors introduce the Neumann version of spherical mean value operators and its generalization. Convolution operators and distributions are defined as well. The next sections are devoted to the invertibility of distributions, the surjectivity of the Neumann mean value operator on the space of smooth functions and some other related topics. Reviewer: Deshna Loonker (Jodhpur) Analysis paths. Volume 1. Schwartz space, tempered distributions and Fourier transformation https://www.zbmath.org/1485.46001 2022-06-24T15:10:38.853281Z "Chiron, David" https://www.zbmath.org/authors/?q=ai:chiron.david Publisher's description: La théorie des distributions, paradigme par excellence, fut révélée au monde par Laurent Schwartz dans les années 1945-50. Elle est le fruit d'une longue maturation (qui a commencé dès le XIXème siècle), où s'illustrent entre autres les noms de O. Heaviside, S. Bochner, P. Dirac, S. Sobolev et J. Leray. Elle a depuis conquis les esprits les plus récalcitrants et a permis en particulier de donner un sens à des calculs ou des opérations indispensables tant en physique mathématique qu'en analyse des équations aux dérivées partielles. La dualité y joue un rôle central. L'invention de l'espace de Schwartz, espace des fonctions infiniment dérivables à décroissance rapide ainsi que leurs dérivées, et par corollaire de l'espace des distributions tempérées, a rendu pertinent aux yeux de tous les opérations de dérivations des distributions et leur implication viscérale dans la théorie de Fourier, le côté frappant de tout cela résidant dans la relative simplicité de la théorie correspondante. Dès lors, l'entrée audacieuse de ces objets mathématiques dans le programme de l'agrégation ne pouvait plus tarder. Le présent livre aborde l'espace de Schwartz, les distributions tempérées et la transformation de Fourier. L'auteur y présuppose de bonnes connaissances sur le calcul différentiel, les espaces de Lebesgue et les convergences dans les espaces fonctionnels, ainsi que la transformation de Fourier pour les fonctions intégrables. Le public visé est donc celui des étudiants de M1 ou de M2. Il est notoire que l'assimilation d'une théorie passe par la pratique d'exercices non triviaux. Dans un style précis et impeccable, David Chiron nous en propose près d'une centaine, de niveaux variés, corrigés avec un soin extrême. Des exemples appropriés sont donnés pour nous familiariser avec les objets en présence : formule des sauts, calculs des dérivées au sens des distributions tempérées, solutions fondamentales des opérateurs usuels (Laplace, von Helmholtz, chaleur, ondes, etc. \dots) et problèmes de Cauchy associés, calculs de transformées de Fourier, théorème d'échantillonnage, théorèmes de Paley-Wiener, etc. Ce premier volume des Chemins d'analyse préfigure déjà, par sa richesse et par le soin apporté à sa finition, l'excellence des livres qui vont suivre. La communauté mathématique en jugera Contributions to mathematics and statistics. Essays in honor of Seppo Hassi https://www.zbmath.org/1485.46002 2022-06-24T15:10:38.853281Z Publisher's description: This Festschrift contains thirteen articles in honor of the sixtieth birthday of Professor Seppo Hassi (University of Vaasa). It centers on three topics: functional analysis and operator theory, boundary value problems, and statistics, stochastics, and the history of mathematics. The collection contains four papers on the topic of functional analysis and operator theory. More precisely, it includes a paper treating the transformation of operator-valued Nevanlinna functions and the congruence of their associated realizing operators, a paper treating Parseval frames in the setting of Krein spaces, a paper treating algebraic inclusions of relations as well as the generalized inverses of relations, and a paper treating Krein-von Neumann and Friedrichs extensions by means of energy spaces. Boundary value problems are considered in six of the contributions. In particular, singular perturbations of the Dirac operator are treated by means of the technique of boundary triplets, the connection between sectorial Schrödinger $$L$$-systems and certain classes of Weyl-Titchmarsh functions is considered, $$PT$$-symmetric Hamiltonians are treated from the perspective of couplings of dual pairs, the Riesz basis property of indefinite Sturm-Liouville problems is considered, the stability properties of spectral characteristics of boundary value problems are investigated, and the completeness and minimality of systems of eigenfunctions and associated functions of ordinary differential operators are treated. Finally, the collection also contains three contributions connected with the topics of statistics, stochastics, and the history of mathematics. More precisely, a new statistic is introduced for the testing of cumulative abnormal returns in the case of partially overlapping event windows, a new characterization of Brownian motion is established, and, finally, a history of (the department of) mathematics and statistics at the University of Vaasa is presented. The articles of this volume will be reviewed individually. Indexed articles: \textit{Arlinskiĭ, Yuri}, Congruence of selfadjoint operators and transformations of operator-valued Nevanlinna functions, 1-14 [Zbl 07523169] \textit{Behrndt, Jussi; Holzmann, Markus; Stelzer, Christian; Stenzel, Georg}, A class of singular perturbations of the Dirac operator: boundary triplets and Weyl functions, 15-35 [Zbl 07523170] \textit{Belyi, Sergey; Tsekanovskiĭ, Eduard}, The original Weyl-Titchmarsh functions and sectorial Schrödinger L-systems, 37-53 [Zbl 07523171] \textit{Derkach, Volodymyr; Schmitz, Philipp; Trunk, Carsten}, $$\mathcal{PT}$$-symmetric Hamiltonians as couplings of dual pairs, 55-68 [Zbl 07523172] \textit{Fleige, Andreas}, Positive and negative examples for the Riesz basis property of indefinite Sturm-Liouville problems, 69-75 [Zbl 07523173] \textit{Kamuda, Alan; Kużel, Sergii}, On Parseval $$J$$-frames, 77-86 [Zbl 07523174] \textit{Labrousse, Jean-Philippe; Sandovici, Adrian; de Snoo, Henk; Winkler, Henrik}, Idempotent relations, semi-projections, and generalized inverses, 87-110 [Zbl 07523175] \textit{Lunyov, Anton; Malamud, Mark}, Lipschitz property of eigenvalues and eigenvectors of $$2\times 2$$ Dirac-type operators, 111-140 [Zbl 07523176] \textit{Möller, Manfred}, Completeness and minimality of eigenfunctions and associated functions of ordinary differential operators, 141-151 [Zbl 07523177] \textit{Pynnönen, Seppo}, Partially overlapping event windows and testing cumulative abnormal returns in financial event studies, 153-164 [Zbl 07523178] \textit{Sebestyén, Zoltán; Tarcsay, Zsigmond}, On the Kreĭn-von Neumann and Friedrichs extension of positive operators, 165-178 [Zbl 07523179] \textit{Sottinen, Tommi}, The characterization of Brownian motion as an isotropic i.i.d.-component Lévy process, 179-186 [Zbl 07523180] \textit{Virtanen, Ilkka}, The role of mathematics and statistics in the University of Vaasa; the first five decades, 187-192 [Zbl 07523181] Seppo Hassi, 60 years, VII-IX [Zbl 07523168] Analysis on Laakso graphs with application to the structure of transportation cost spaces https://www.zbmath.org/1485.46011 2022-06-24T15:10:38.853281Z "Dilworth, S. J." https://www.zbmath.org/authors/?q=ai:dilworth.stephen-j "Kutzarova, Denka" https://www.zbmath.org/authors/?q=ai:kutzarova.denka-n "Ostrovskii, Mikhail I." https://www.zbmath.org/authors/?q=ai:ostrovskii.mikhail-i The authors continue the study of the geometry of transportation cost spaces (also called Arens-Eells spaces and Lipschitz-free spaces) on certain finite metric spaces they initiated in [\textit{S.~J. Dilworth} et al., Can. J. Math. 72, No.~3, 774-804 (2020; Zbl 1462.46010)]. They focus on Laakso graphs $$L_n$$ and multi-branching diamonds $$D_n^k$$; these are relevant families since their bi-Lipschitz embeddability characterizes non-superreflexive Banach spaces. The authors prove a lower bound for the Banach-Mazur distance from the transportation cost space on Laakso graphs to $$\ell_1^N$$. To this end, they construct orthogonal bases for the cycle and cut spaces and use them to estimate the projection constant of $$\operatorname{Lip}_0(L_n)$$. The authors also calculate the exact projection constant of $$\operatorname{Lip}_0(D_{n,k})$$ and deduce a lower bound for the Banach-Mazur distance to $$\ell_1^N$$ that improves the one they obtained in [loc. cit.]. In the last section, examples of finite metric spaces whose transportation cost spaces contain $$\ell_\infty^3$$ and $$\ell_\infty^4$$ isometrically are provided (we refer the reader to [\textit{M.~Alexander} et al. J. Funct. Anal. 280, No.~4, Article ID 108849, 39~p. (2021; Zbl 1459.52007)] for an example isometric to~$$\ell_\infty^3$$). Reviewer: Luis C. García Lirola (Zaragoza) Optimal approximants and orthogonal polynomials in several variables. II: Families of polynomials in the unit ball https://www.zbmath.org/1485.46030 2022-06-24T15:10:38.853281Z "Sargent, Meredith" https://www.zbmath.org/authors/?q=ai:sargent.meredith "Sola, Alan A." https://www.zbmath.org/authors/?q=ai:sola.alan-a This paper is devoted to a specific case of Hilbert spaces of functions of two complex variables. The authors introduce certain weighted orthogonal polynomials and study optimal approximants associated with the function $$f(z)=1-\frac{1}{\sqrt{2}}(z_1 +z_2)$$, and they find a scale of Hilbert function spaces in the unit 2-ball $\mathbb B^2$ with the reproducing kernel of the form $$(1-\langle z,w\rangle)^{-\gamma}$$, where $$\gamma > 0$$. Reviewer: Saeed Hashemi Sababe (Tehran) A characterization of spaces of homogeneous type induced by continuous ellipsoid covers of $$\mathbb{R}^n$$ https://www.zbmath.org/1485.46032 2022-06-24T15:10:38.853281Z "Bownik, Marcin" https://www.zbmath.org/authors/?q=ai:bownik.marcin "Li, Baode" https://www.zbmath.org/authors/?q=ai:li.baode "Weissblat, Tal" https://www.zbmath.org/authors/?q=ai:weissblat.tal Summary: We study the relationship between the concept of a continuous ellipsoid $$\Theta$$ cover of $$\mathbb{R}^n$$, which was introduced by \textit{W. Dahmen} et al. [Numer. Math. 107, No. 3, 503--532 (2007; Zbl 1129.65092), Constr. Approx. 31, No. 2, 149--194 (2010; Zbl 1195.46030)], (see also \textit{S. Dekel} and \textit{P. Petrushev} [in: Multiscale, nonlinear and adaptive approximation. Dedicated to Wolfgang Dahmen on the occasion of his 60th birthday. Berlin: Springer. 137--167 (2009; Zbl 1196.46019)]), and the space of homogeneous type induced by $$\Theta$$. We characterize the class of quasi-distances on $$\mathbb{R}^n$$ (up to equivalence) which correspond to continuous ellipsoid covers. This places firmly continuous ellipsoid covers as a subclass of spaces of homogeneous type on $$\mathbb{R}^n$$ satisfying quasi-convexity and 1-Ahlfors-regularity. Divergent Fourier series in function spaces near $$L^1[0;1]$$ https://www.zbmath.org/1485.46033 2022-06-24T15:10:38.853281Z "Kopaliani, Tengiz" https://www.zbmath.org/authors/?q=ai:kopaliani.tengiz "Samashvili, Nino" https://www.zbmath.org/authors/?q=ai:samashvili.nino "Zviadadze, Shalva" https://www.zbmath.org/authors/?q=ai:zviadadze.shalva Summary: Bochkariev's theorem states that for any uniformly bounded orthonormal system $$\Phi$$, there is a Lebesgue integrable function such that the Fourier series of it with respect to the system $$\Phi$$ diverges on the set of positive measure. In this paper, we extended Bochkariev's theorem for some class of variable exponent Lebesgue spaces. We characterized the class of variable exponent Lebesgue spaces $$L^{p ( \cdot )} [0; 1]$$, $$1 < p(x) < \infty$$ a.e. on $$[0;1]$$, such that above mentioned Bochkarev's theorem is valid. Real interpolation of martingale Orlicz Hardy spaces and BMO spaces https://www.zbmath.org/1485.46034 2022-06-24T15:10:38.853281Z "Long, Long" https://www.zbmath.org/authors/?q=ai:long.long "Weisz, Ferenc" https://www.zbmath.org/authors/?q=ai:weisz.ferenc "Xie, Guangheng" https://www.zbmath.org/authors/?q=ai:xie.guangheng Summary: In this article, the authors prove that the real interpolation spaces between martingale Orlicz Hardy spaces and martingale BMO spaces are martingale Orlicz-Lorentz Hardy spaces. Using sharp maximal functions, the authors also establish the characterizations of martingale Orlicz Hardy spaces. Szasz's theorem and its generalizations https://www.zbmath.org/1485.46036 2022-06-24T15:10:38.853281Z "Bourdaud, Gérard" https://www.zbmath.org/authors/?q=ai:bourdaud.gerard The author investigates generalizations of a theorem of Szasz, given by the estimate $\int_{{\mathbb{R}^n}} |\xi|^{\theta p} |\mathcal{F} f(\xi)|^p d\xi \le c \, \|f\|_{\dot{A}^s_{r,q}{\mathbb{R}^n}}\, ,$ where $$\mathcal{F}$$ is the Fourier transform, $$c$$ a general constant depending only on the parameters $$s,p,q,r,n, \theta$$ and $$\dot{A}^s_{r,q}{\mathbb{R}^n}$$ denotes either the homogeneous Besov space $$\dot{B}^s_{r,q}(\mathbb{R}^n)$$ or the homogeneous Triebel-Lizorkin space $$\dot{F}^s_{r,q}(\mathbb{R}^n)$$. His main result consists in an if and only if assertion for the validity of the above inequality. Reviewer: Winfried Sickel (Jena) Weighted Besov spaces with variable exponents https://www.zbmath.org/1485.46043 2022-06-24T15:10:38.853281Z "Wang, Shengrong" https://www.zbmath.org/authors/?q=ai:wang.shengrong "Xu, Jingshi" https://www.zbmath.org/authors/?q=ai:xu.jingshi Summary: In this paper, we introduce Besov spaces with variable exponents and variable Muckenhoupt weights. Then we give a approximation characterization, the lifting property, embeddings, the duality and interpolation of these spaces. Singular integrals, rank one perturbations and Clark model in general situation https://www.zbmath.org/1485.47016 2022-06-24T15:10:38.853281Z "Liaw, Constanze" https://www.zbmath.org/authors/?q=ai:liaw.constanze "Treil, Sergei" https://www.zbmath.org/authors/?q=ai:treil.sergei Summary: We start with considering rank one self-adjoint perturbations $$A_{\alpha}=A+ \alpha (\cdot, \varphi) \varphi$$ with cyclic vector $$\varphi \in \mathcal{H}$$ on a separable Hilbert space $$\mathcal{H}$$. The spectral representation of the perturbed operator $$A_{\alpha}$$ is realized by a (unitary) operator of a special type: the Hilbert transform in the two-weight setting, the weights being spectral measures of the operators $$A$$ and $$A_{\alpha}$$. Similar results will be presented for unitary rank one perturbations of unitary operators, leading to singular integral operators on the circle. This motivates the study of abstract singular integral operators, in particular the regularization of such operator in very general settings. Further, starting with contractive rank one perturbations we present the Clark theory for arbitrary spectral measures (i.e. for arbitrary, possibly not inner characteristic functions). We present a description of the Clark operator and its adjoint in the general settings. Singular integral operators, in particular the so-called normalized Cauchy transform again plays a prominent role. Finally, we present a possible way to construct the Clark theory for dissipative rank one perturbations of self-adjoint operators. These lecture notes give an account of the mini-course delivered by the authors, which was centered around [\textit{C. Liaw} and \textit{S. Treil}, J. Funct. Anal. 257, No. 6, 1947--1975 (2009; Zbl 1206.42012); Rev. Mat. Iberoam. 29, No. 1, 53--74 (2013; Zbl 1272.42011); J. Anal. Math. 130, 287--328 (2016; Zbl 06697868)]. Unpublished results are restricted to the last part of this manuscript. For the entire collection see [Zbl 1381.00044]. Approximation of mixed order Sobolev functions on the $$d$$-torus: asymptotics, preasymptotics, and $$d$$-dependence https://www.zbmath.org/1485.47025 2022-06-24T15:10:38.853281Z "Kühn, Thomas" https://www.zbmath.org/authors/?q=ai:kuhn.thomas "Sickel, Winfried" https://www.zbmath.org/authors/?q=ai:sickel.winfried "Ullrich, Tino" https://www.zbmath.org/authors/?q=ai:ullrich.tino Summary: We investigate the approximation of $$d$$-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness $$s>0$$ on the $$d$$-dimensional torus, where the approximation error is measured in the $$L_2$$-norm. In other words, we study the approximation numbers $$a_n$$ of the Sobolev embeddings $$H^s_{\mathrm{mix}}(\mathbb{T}^d)\hookrightarrow L_2(\mathbb{T}^d)$$, with particular emphasis on the dependence on the dimension $$d$$. For any fixed smoothness $$s>0$$, we find two-sided estimates for the approximation numbers as a function in $$n$$ and $$d$$. We observe super-exponential decay of the constants in $$d$$, if $$n$$, the number of linear samples of $$f$$, is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by approximations using only a few linear samples (small $$n$$). We present some surprising results for the so-called preasymptotic'' decay and point out connections to the recently introduced notion of quasi-polynomial tractability of approximation problems. Sparse domination of weighted composition operators on weighted Bergman spaces https://www.zbmath.org/1485.47036 2022-06-24T15:10:38.853281Z "Hu, Bingyang" https://www.zbmath.org/authors/?q=ai:hu.bingyang "Li, Songxiao" https://www.zbmath.org/authors/?q=ai:li.songxiao "Shi, Yecheng" https://www.zbmath.org/authors/?q=ai:shi.yecheng "Wick, Brett D." https://www.zbmath.org/authors/?q=ai:wick.brett-d The paper uses the technique of sparse domination from harmonic analysis to study several problems for the (holomorphic) Bergman spaces of the upper-half plane and the open unit ball. The problems studied include Carleson embedding, the boundedness and compactness of weighted composition operators, and weighted type estimates for functions in the holomorphic Bergman spaces. Reviewer: Kehe Zhu (Albany) Self-affine tiling of polyhedra https://www.zbmath.org/1485.52015 2022-06-24T15:10:38.853281Z "Protasov, V. Yu." https://www.zbmath.org/authors/?q=ai:protasov.vladimir-yu "Zaitseva, T. I." https://www.zbmath.org/authors/?q=ai:zaitseva.t-i Summary: We obtain a complete classification of polyhedral sets (unions of finitely many convex polyhedra) that admit self-affine tilings, i.e., partitions into parallel shifts of one set that is affinely similar to the initial one. In every dimension, there exist infinitely many nonequivalent polyhedral sets possessing this property. Under an additional assumption that the affine similarity is defined by an integer matrix and by integer shifts (digits'') from different quotient classes with respect to this matrix, the only polyhedral set of this kind is a parallelepiped. Applications to multivariate wavelets and to Haar systems are discussed. Optimization of neural network training for image recognition based on trigonometric polynomial approximation https://www.zbmath.org/1485.68239 2022-06-24T15:10:38.853281Z "Vershkov, N." https://www.zbmath.org/authors/?q=ai:vershkov.n "Babenko, M." https://www.zbmath.org/authors/?q=ai:babenko.mikhail "Tchernykh, A." https://www.zbmath.org/authors/?q=ai:tchernykh.andrei "Pulido-Gaytan, B." https://www.zbmath.org/authors/?q=ai:pulido-gaytan.b "Cortés-Mendoza, J. M." https://www.zbmath.org/authors/?q=ai:cortes-mendoza.jorge-m "Kuchukov, V." https://www.zbmath.org/authors/?q=ai:kuchukov.viktor "Kuchukova, N." https://www.zbmath.org/authors/?q=ai:kuchukova.n Summary: The paper discusses optimization issues of training Artificial Neural Networks (ANNs) using a nonlinear trigonometric polynomial function. The proposed method presents the mathematical model of an ANN as an information transmission system where effective techniques to restore signals are widely used. To optimize ANN training, we use energy characteristics assuming ANNs as data transmission systems. We propose a nonlinear layer in the form of a trigonometric polynomial that approximates the syncular'' function based on the generalized approximation theorem and the wave model. To confirm the theoretical results, the efficiency of the proposed approach is compared with standard ANN implementations with sigmoid and Rectified Linear Unit (ReLU) activation functions. The experimental evaluation shows the same accuracy of standard ANNs with a time reduction of the training phase of supervised learning for the proposed model. Risk related prediction for recurrent stroke and post-stroke epilepsy using fractional Fourier transform analysis of EEG signals https://www.zbmath.org/1485.92059 2022-06-24T15:10:38.853281Z "Dulf, Eva-H." https://www.zbmath.org/authors/?q=ai:dulf.eva-henrietta "Ionescu, Clara-M." https://www.zbmath.org/authors/?q=ai:ionescu.clara-mihaela Summary: Stroke is a medical condition which can easily affect the quality of life, depending on how extended the stroke is and what regions of the brain are involved. According to the most recent data cited in WHO, Romania is in top three of the countries with increased frequency of stroke and has the second place for having the most deaths and disabilities caused by stroke. Actually, stroke is the second death cause in Romania after cardiac arrest. Today, there are various prevention methods concerning stroke. The hypothesis of the research context is that EEG signal can provide useful information on risk related prediction for recurrent stroke and post-stroke epilepsy. Knowing that there is a certain risk on developing secondary epilepsy after stroke, based on the EEG rhythms, may help in prevention and maybe in reconsidering a new approach in the treatment of this pathology. On the other hand fractional Fourier transform (FrFT), a generalization of conventional Fourier transform, is used with success in many applications like detection of signals in noise, image compression, reduction of side lobe levels using convolutional windows, time-frequency analysis, etc. It can be used in more effective manner compared to Fourier transform with additional degrees of freedom. That was the motivation to analyze the spectra of each component of the EEG signals using FrFT in order to predict recurrent stroke and post-stroke epilepsy incidence. The results prove the efficiency of the method. For the entire collection see [Zbl 1477.93110]. Uncertainty principles associated to sets satisfying the geometric control condition https://www.zbmath.org/1485.93121 2022-06-24T15:10:38.853281Z "Green, Walton" https://www.zbmath.org/authors/?q=ai:green.walton "Jaye, Benjamin" https://www.zbmath.org/authors/?q=ai:jaye.benjamin-j "Mitkovski, Mishko" https://www.zbmath.org/authors/?q=ai:mitkovski.mishko Summary: In this paper, we study forms of the uncertainty principle suggested by problems in control theory. We obtain a version of the classical Paneah-Logvinenko-Sereda theorem for the annulus. More precisely, we show that a function with spectrum in an annulus of a given thickness can be bounded, in $$L^2$$-norm, from above by its restriction to a neighborhood of a GCC set, with constant independent of the radius of the annulus. We apply this result to obtain energy decay rates for damped fractional wave equations, extending the work of Malhi and Stanislavova to both the higher-dimensional and non-periodic setting. Differential pulse code modulation and motion aligned optimal reconstruction for block-based compressive video sensing using conditional autoregressive-salp swarm algorithm https://www.zbmath.org/1485.94010 2022-06-24T15:10:38.853281Z "Sekar, R." https://www.zbmath.org/authors/?q=ai:sekar.r-chandra-guru|sekar.ramamurthy|sekar.rajam "Ravi, G." https://www.zbmath.org/authors/?q=ai:ravi.g Geometry of the phase retrieval problem. Graveyard of algorithms https://www.zbmath.org/1485.94013 2022-06-24T15:10:38.853281Z "Barnett, Alexander H." https://www.zbmath.org/authors/?q=ai:barnett.alexander-h "Epstein, Charles L." https://www.zbmath.org/authors/?q=ai:epstein.charles-l "Greengard, Leslie" https://www.zbmath.org/authors/?q=ai:greengard.leslie-f "Magland, Jeremy" https://www.zbmath.org/authors/?q=ai:magland.jeremy-f Publisher's description: Recovering the phase of the Fourier transform is a ubiquitous problem in imaging applications from astronomy to nanoscale X-ray diffraction imaging. Despite the efforts of a multitude of scientists, from astronomers to mathematicians, there is, as yet, no satisfactory theoretical or algorithmic solution to this class of problems. Written for mathematicians, physicists and engineers working in image analysis and reconstruction, this book introduces a conceptual, geometric framework for the analysis of these problems, leading to a deeper understanding of the essential, algorithmically independent, difficulty of their solutions. Using this framework, the book studies standard algorithms and a range of theoretical issues in phase retrieval and provides several new algorithms and approaches to this problem with the potential to improve the reconstructed images. The book is lavishly illustrated with the results of numerous numerical experiments that motivate the theoretical development and place it in the context of practical applications. Wigner analysis of operators. I: Pseudodifferential operators and wave fronts https://www.zbmath.org/1485.94017 2022-06-24T15:10:38.853281Z "Cordero, Elena" https://www.zbmath.org/authors/?q=ai:cordero.elena "Rodino, Luigi" https://www.zbmath.org/authors/?q=ai:rodino.luigi Summary: We perform Wigner analysis of linear operators. Namely, the standard time-frequency representation Short-time Fourier Transform (STFT) is replaced by the $$\mathcal{A}$$-Wigner distribution defined by $$W_{\mathcal{A}}(f) = \mu(\mathcal{A})(f \otimes \bar{f})$$, where $$\mathcal{A}$$ is a $$4 d \times 4 d$$ symplectic matrix and $$\mu(\mathcal{A})$$ is an associate metaplectic operator. Basic examples are given by the so-called $$\tau$$-Wigner distributions. Such representations provide a new characterization for modulation spaces when $$\tau \in(0, 1)$$. Furthermore, they can be efficiently employed in the study of the off-diagonal decay for pseudodifferential operators with symbols in the Sjöstrand class (in particular, in the Hörmander class $$S_{0 , 0}^0)$$. The novelty relies on defining time-frequency representations via metaplectic operators, developing a conceptual framework and paving the way for a new understanding of quantization procedures. We deduce micro-local properties for pseudodifferential operators in terms of the Wigner wave front set. Finally, we compare the Wigner with the global Hörmander wave front set and identify the possible presence of a ghost region in the Wigner wave front. In the second part of the paper applications to Fourier integral operators and Schrödinger equations will be given. Uncertainty principles for the two-sided quaternion linear canonical transform https://www.zbmath.org/1485.94032 2022-06-24T15:10:38.853281Z "Zhu, Xiaoyu" https://www.zbmath.org/authors/?q=ai:zhu.xiaoyu "Zheng, Shenzhou" https://www.zbmath.org/authors/?q=ai:zheng.shenzhou Summary: The quaternion linear canonical transform (QLCT), as a generalized form of the quaternion Fourier transform, is a powerful analyzing tool in image and signal processing. In this paper, we propose three different forms of uncertainty principles for the two-sided QLCT, which include Hardy's uncertainty principle, Beurling's uncertainty principle and Donoho-Stark's uncertainty principle. These consequences actually describe the quantitative relationships of the quaternion-valued signal in arbitrary two different QLCT domains, which have many applications in signal recovery and color image analysis. In addition, in order to analyze the non-stationary signal and time-varying system, we present Lieb's uncertainty principle for the two-sided short-time quaternion linear canonical transform (SQLCT) based on the Hausdorff-Young inequality. By adding the nonzero quaternion-valued window function, the two-sided SQLCT has a great significant application in the study of signal local frequency spectrum. Finally, we also give a lower bound for the essential support of the two-sided SQLCT.