Recent zbMATH articles in MSC 41Ahttps://www.zbmath.org/atom/cc/41A2021-04-16T16:22:00+00:00WerkzeugOn reconstruction of dynamic permeability and tortuosity from data at distinct frequencies.https://www.zbmath.org/1456.740372021-04-16T16:22:00+00:00"Ou, Miao-Jung Yvonne"https://www.zbmath.org/authors/?q=ai:ou.miao-jung-yvonneA continuant and an estimate of the remainder of the interpolating continued \(C\)-fraction.https://www.zbmath.org/1456.300082021-04-16T16:22:00+00:00"Pahirya, M. M."https://www.zbmath.org/authors/?q=ai:pahirya.mykhaylo-m|pagirya.m-mSummary: The problem of the interpolation of functions of a real variable by interpolating continued \(C\)-fraction is investigated. The relationship between the continued fraction and the continuant was used. The properties of the continuant are established. The formula for the remainder of the interpolating continued \(C\)-fraction proved. The remainder expressed in terms of derivatives of the functional continent. An estimate of the remainder was obtained. The main result of this paper is contained in the following Theorem 5:
Let \(\mathcal{R}\subset \mathbb{R}\) be a compact, \(f \in \mathbf{C}^{(n+1)}(\mathcal{R})\) and the interpolating continued \(C\)-fraction (\(C\)-ICF) of the form
\[D_n(x)=\frac{P_n(x)}{Q_n(x)}=a_0+\frac{K}{k=1}{n}\frac{a_k(x-x_{k-1})}{1}, a_k \in \mathbb{R}, \; k=\overline{0,n},\]
be constructed by the values the function \(f\) at nodes \(X=\{x_i : x_i \in \mathcal{R}\), \(x_i\neq x_j\), \(i\neq j\), \(i,j=\overline{0,n}\}\). If the partial numerators of \(C\)-ICF satisfy the condition of the Paydon-Wall type, that is \(0<a^* \operatorname{diam} \mathcal{R} \leq p\), then
\[\begin{aligned} |f(x)-D_n(x)|\leq \frac{f^*\prod\limits_{k=0}^n |x-x_k|}{(n+1)! \Omega_n(t)} \Bigg( \kappa_{n+1}(p)+\sum_{k=1}^r \binom{n+1}{k} (a^*)^k \sum_{i_1=1}^{n+1-2k} \kappa_{i_1}(p)\times \\
\times \sum_{i_2=i_1+2}^{n+3-3k} \kappa_{i_2-i_1-1}(p)\dots \sum_{i_{k-1}=i_{k-2}+2}^{n-3} \kappa_{i_{k-1}-i_{k-2}-1}(p) \sum_{i_k=i_{k-1}+2}^{n-1} \kappa_{i_k-i_{k-1}-1}(p) \kappa_{n-i_k}(p)\Bigg),\end{aligned}\]
where \(f^*= \max\limits_{0\leq m \leq r}\max\limits_{x \in \mathcal{R}} |f^{(n+1-m)}(x)|\), \( \kappa_n(p)=\frac{(1+\sqrt{1+4p})^n-(1-\sqrt{1+4p})^n}{2^n \sqrt{1+4p}}\), \(a^*=\max\limits_{2\leqslant i \leqslant n}|a_i|\), \(p=t(1-t)\), \(t\in(0;\frac{1}{2}]\), \(r=\big[\frac{n}{2}\big]\).Regularity of the solution to fractional diffusion, advection, reaction equations in weighted Sobolev spaces.https://www.zbmath.org/1456.352122021-04-16T16:22:00+00:00"Ervin, V. J."https://www.zbmath.org/authors/?q=ai:ervin.vincent-jSummary: In this article we investigate the regularity of the solution to the fractional diffusion, advection, reaction equation on a bounded domain in \(\mathbb{R}^1\). The analysis is performed in the weighted Sobolev spaces, \( H_{(a, b)}^s(\text{I})\). Three different characterizations of \(H_{(a, b)}^s (\text{I})\) are presented, together with needed embedding theorems for these spaces. The analysis shows that the regularity of the solution is bounded by the endpoint behavior of the solution, which is determined by the parameters \(\alpha\) and \(r\) defining the fractional diffusion operator. Additionally, the analysis shows that for a sufficiently smooth right hand side function, the regularity of the solution to fractional diffusion reaction equation is lower than that of the fractional diffusion equation. Also, the regularity of the solution to fractional diffusion advection reaction equation is two orders lower than that of the fractional diffusion reaction equation.Approximation by sub-matrix means of multiple Fourier series in the Hölder metric.https://www.zbmath.org/1456.420102021-04-16T16:22:00+00:00"Krasniqi, Xhevat Z."https://www.zbmath.org/authors/?q=ai:krasniqi.xhevat-zahir|krasniqi.xhevat-zSummary: In this paper some results on approximation by sub-matrix means of multiple Fourier series in the Hölder metric are obtained. Our results are applicable for a wider class of sequences and give a better degree of approximation than those presented previously by others.Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems.https://www.zbmath.org/1456.828042021-04-16T16:22:00+00:00"Reisinger, Christoph"https://www.zbmath.org/authors/?q=ai:reisinger.christoph"Zhang, Yufei"https://www.zbmath.org/authors/?q=ai:zhang.yufeiThis is an investigation on the possibility to approximate nonsmooth value functions associated to stiff stochastic differential equations (SDEs), by means of deep artificial neural networks (DNNs) with polynomial complexity. The intention is to prove that DNNs can overcome the curse of dimensionality for approximating value functions of zero-sum games of controlled SDEs with stiff, time-inhomogeneous nonlinear coefficients. One shows that the viscosity solution to a Kolmogorov backward partial differential equation with stiff coefficients can be approximated by DNNs with polynomial complexity. The second section of the paper is devoted to the introduction of DNN, d-dimensional controlled SDE and the presentation of main results of the paper. Under some specific assumptions, one presents the expression rate of DNNs for approximating value functions induced by nonlinear SDEs with stiff coefficients and the expression rate of DNNs with polynomial complexity to approximate value functions associated with a sequence of controlled SDEs with stiff coefficients. Basic operations to construct DNNs from existing ones and fundamental results on the representation flexibility of DNNs, the so-called Rectified Linear Unit (Re LU) Network Calculus, are discussed in the third section. Error estimates of linear-implicit Euler discretization for a finite-dimensional SDE are shown in the fourth section. The convergence analysis results in this section are extended in the next section to SDEs controlled by a piecewise-constant deterministic strategy. The sixth and seventh sections are devoted to the proofs of the main results. A short discussion of the results is presented in the eighth section and Appendix A collects few basic operations of Re LU DNNs.
Reviewer: Claudia Simionescu-Badea (Wien)On a Hilbert-type inequality with homogeneous kernel involving hyperbolic functions.https://www.zbmath.org/1456.260212021-04-16T16:22:00+00:00"You, Minghui"https://www.zbmath.org/authors/?q=ai:you.minghui"Guan, Yue"https://www.zbmath.org/authors/?q=ai:guan.yueSeveral Hilbert-type integral inequalities were established in the last years, with best constant factors, different new kernel functions, special constants such as Bernoulli numbers and Euler numbers and containing special functions. Here, the authors construct a homogeneous kernel function involving hyperbolic functions and establish a Hilbert-type inequality and its equivalent Hardy form with best possible constant factors. Furthermore, they introduce Bernoulli numbers and the partial fraction expansions of trigonometric functions, and then present several Hilbert-type inequalities in which the constant factors are represented by Bernoulli numbers and by some higher-order derivatives of trigonometric functions. Several corollaries and examples illustrate the general results.
Reviewer: Ioan Raşa (Cluj-Napoca)On filtered polynomial approximation on the sphere.https://www.zbmath.org/1456.410062021-04-16T16:22:00+00:00"Wang, Heping"https://www.zbmath.org/authors/?q=ai:wang.heping"Sloan, Ian H."https://www.zbmath.org/authors/?q=ai:sloan.ian-hSummary: This paper considers filtered polynomial approximations on the unit sphere \(\mathbb {S}^d\subset \mathbb {R}^{d+1}\), obtained by truncating smoothly the Fourier series of an integrable function \(f\) with the help of a ``filter'' \(h\), which is a real-valued continuous function on \([0,\infty)\) such that \(h(t)=1\) for \(t\in [0,1]\) and \(h(t)=0\) for \(t\geq 2\). The resulting ``filtered polynomial approximation'' (a spherical polynomial of degree \(2L-1\)) is then made fully discrete by approximating the inner product integrals by an \(N\)-point cubature rule of suitably high polynomial degree of precision, giving an approximation called ``filtered hyperinterpolation''. In this paper we require that the filter \(h\) and all its derivatives up to \(\lfloor \tfrac{d-1}{2}\rfloor \) are absolutely continuous, while its right and left derivatives of order \(\lfloor \tfrac{d+1}{2}\rfloor \) exist everywhere and are of bounded variation. Under this assumption we show that for a function \(f\) in the Sobolev space \(W^s_p(\mathbb {S}^d),\;1\leq p\leq \infty \), both approximations are of the optimal order \( L^{-s}\), in the first case for \(s>0\) and in the second fully discrete case for \(s>d/p\), conditions which in both cases cannot be weakened.A generalized fractional-order Chebyshev wavelet method for two-dimensional distributed-order fractional differential equations.https://www.zbmath.org/1456.651302021-04-16T16:22:00+00:00"Do, Quan H."https://www.zbmath.org/authors/?q=ai:do.quan-h"Ngo, Hoa T. B."https://www.zbmath.org/authors/?q=ai:ngo.hoa-t-b"Razzaghi, Mohsen"https://www.zbmath.org/authors/?q=ai:razzaghi.mohsenSummary: We provide a new effective method for the two-dimensional distributed-order fractional differential equations (DOFDEs). The technique is based on fractional-order Chebyshev wavelets. An exact formula involving regularized beta functions for determining the Riemann-Liouville fractional integral operator of these wavelets is given. The given wavelets and this formula are utilized to find the solutions of the given two-dimensional DOFDEs. The method gives very accurate results. The given numerical examples support this claim.On circumcenters of finite sets in Hilbert spaces.https://www.zbmath.org/1456.510082021-04-16T16:22:00+00:00"Bauschke, Heinz H."https://www.zbmath.org/authors/?q=ai:bauschke.heinz-h"Ouyang, Hui"https://www.zbmath.org/authors/?q=ai:ouyang.hui"Wang, Xianfu"https://www.zbmath.org/authors/?q=ai:wang.xianfuSummary: A well-known object in classical Euclidean geometry is the circumcenter of a triangle, i.e., the point that is equidistant from all vertices. The purpose of this paper is to provide a systematic study of the circumcenter of sets containing finitely many points in a Hilbert space. This is motivated by recent works of \textit{R. Behling} et al. [Numer. Algorithms 78, No. 3, 759--776 (2018; Zbl 1395.49023); Oper. Res. Lett. 46, No. 2, 159--162 (2018; Zbl 07064464)] on accelerated versions of the Douglas-Rachford method. We present basic results and properties of the circumcenter. Several examples are provided to illustrate the tightness of various assumptions.Coproximinality in spaces of Bochner integrable functions.https://www.zbmath.org/1456.410082021-04-16T16:22:00+00:00"Rao, T. S. S. R. K."https://www.zbmath.org/authors/?q=ai:rao.t-s-s-r-kSummary: In this short note we use a new way of applying von Neumann's selection theorem for obtaining best coapproximation in spaces of measurable functions. For a coproximinal closed subspace \(Y\) of a Banach space \(X\), we show that if \(Y\) is constrained in a weakly compactly generated dual space, then the space \(L^1(\mu,Y)\) of \(Y\)-valued Bochner integrable functions is coproximinal in \(L^1(\mu,X)\). This extends a result of Haddadi et al., proved when \(Y\) is reflexive.Approximation by gamma type operators.https://www.zbmath.org/1456.410042021-04-16T16:22:00+00:00"Deveci, Serife Nur"https://www.zbmath.org/authors/?q=ai:deveci.serife-nur"Acar, Tuncer"https://www.zbmath.org/authors/?q=ai:acar.tuncer"Alagoz, Osman"https://www.zbmath.org/authors/?q=ai:alagoz.osmanA new version of the so-called gamma type operators in the theory of approximation by positive operators is introduced, which maps a function \(e^{2\mu x}\) into itself. The classical analogs of the corresponding results for these operators are obtained. A numerical implementation is carried out and the rates of convergence are compared.
Reviewer: Bilal Bilalov (Baku)A projection algorithm on the set of polynomials with two bounds.https://www.zbmath.org/1456.650122021-04-16T16:22:00+00:00"Campos Pinto, M."https://www.zbmath.org/authors/?q=ai:campos-pinto.martin"Charles, F."https://www.zbmath.org/authors/?q=ai:charles.frederique"Després, B."https://www.zbmath.org/authors/?q=ai:despres.bruno"Herda, M."https://www.zbmath.org/authors/?q=ai:herda.maximeSummary: The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in [\textit{B. Després}, Numer. Algorithms 76, No. 3, 829--859 (2017; Zbl 1379.65010)] and [\textit{B. Despres} and \textit{M. Herda}, Numer. Algorithms 77, No. 1, 309--311 (2018; Zbl 1381.65009)] where an interpretation of monovariate polynomials with two bounds is provided in terms of a quaternion algebra and the Euler four-squares formulas. Thanks to this structure, we generate a new nonlinear projection algorithm onto the set of polynomials with two bounds. The numerical analysis of the method provides theoretical error estimates showing stability and continuity of the projection. Some numerical tests illustrate this novel algorithm for constrained polynomial approximation.Sharp Remez inequality.https://www.zbmath.org/1456.410022021-04-16T16:22:00+00:00"Tikhonov, S."https://www.zbmath.org/authors/?q=ai:tikhonov.sergei-vladimirovich|tikhonov.sergei-viktorovich.1|tikhonov.s-n|tikhonov.sergey-yu|tikhonov.sergei-viktorovich"Yuditskii, P."https://www.zbmath.org/authors/?q=ai:yuditskii.peterThe classical Remez inequality asserts that for any polynomial \(P_n\) of degree \(n\) with \linebreak
\(\big|\bigl\{x\in[-1,1]:|P_n(x)|\leq1\bigr\}\big|\geq2-s\), \(0 < s < 2\), the inequality
\[
\max_{x\in[-1,1]}|P_n(x)|\leq T_n\bigl(\tfrac{2+s}{2-s}\bigr)
\]
holds, where \(|B|\) denotes the Lebesgue measure of a measurable set \(B\) and \(T_n(x)=\cos(n\arccos(x))\) is the classical Chebyshev polynomial of the first kind. In this paper, the authors prove a similar inequality for sets on the unit circle \({\mathbb T}\), that is, for any polynomial \(P_n\) of degree \(n\) with \(\big|\bigl\{z\in{\mathbb T}:|P_n(z)|\leq1\bigr\}\big|\geq2\pi-s\), \(0 < s < 2\pi\), the inequality
\[
\max_{z\in{\mathbb T}}|P_n(z)|\leq T_n(1/\cos(\tfrac{s}{4}))
\]
holds.
Reviewer: Klaus Schiefermayr (Wels)Multilevel interpolation of scattered data using \(\mathcal{H}\)-matrices.https://www.zbmath.org/1456.650112021-04-16T16:22:00+00:00"Le Borne, Sabine"https://www.zbmath.org/authors/?q=ai:le-borne.sabine"Wende, Michael"https://www.zbmath.org/authors/?q=ai:wende.michaelSummary: Scattered data interpolation can be used to approximate a multivariate function by a linear combination of positive definite radial basis functions (RBFs). In practice, the approximation error stagnates (due to numerical instability) even if the function is smooth and the number of data centers is increased. A smaller approximation error can be obtained using multilevel interpolation on a sequence of nested subsets of the initial set of centers. For the construction of these nested subsets, we compare two thinning algorithms from the literature, a greedy algorithm based on nearest neighbor computations and a Poisson point process. The main novelty of our approach lies in the use of \(\mathcal{H}\)-matrices both for the solution of linear systems and for the evaluation of residual errors at each level. For the solution of linear systems, we use GMRes combined with a domain decomposition preconditioner. Using \(\mathcal{H}\)-matrices allows us to solve larger problems more efficiently compared with multilevel interpolation based on dense matrices. Numerical experiments with up to 50,000 scattered centers in two and three spatial dimensions demonstrate that the computational time required for the construction of the multilevel interpolant using \(\mathcal{H}\)-matrices is of almost linear complexity with respect to the number of centers.Strongly quasinonexpansive mappings. II.https://www.zbmath.org/1456.470162021-04-16T16:22:00+00:00"Aoyama, Koji"https://www.zbmath.org/authors/?q=ai:aoyama.koji"Zembayashi, Kei"https://www.zbmath.org/authors/?q=ai:zembayashi.keiSummary: This paper is devoted to the study of strongly quasinonexpansive mappings in an abstract space and a Banach space.
Editorial remark. Part I has appeared in [\textit{K. Aoyama}, in: Proceedings of the 9th International Conference on Nonlinear analysis and convex analysis, Yokohama: Yokohama Publ. 19--27 (2016; per bibl.)]. Part III has appeared in [\textit{K. Aoyama} and \textit{F. Kohsaka}, Linear Nonlinear Anal. 6, No. 1, 1--12 (2020), \url{http://www.yokohamapublishers.jp/online2/oplna/vol6/p1.html}].Fast component-by-component construction of lattice algorithms for multivariate approximation with POD and SPOD weights.https://www.zbmath.org/1456.650162021-04-16T16:22:00+00:00"Cools, Ronald"https://www.zbmath.org/authors/?q=ai:cools.ronald"Kuo, Frances Y."https://www.zbmath.org/authors/?q=ai:kuo.frances-y"Nuyens, Dirk"https://www.zbmath.org/authors/?q=ai:nuyens.dirk"Sloan, Ian H."https://www.zbmath.org/authors/?q=ai:sloan.ian-hSummary: In a recent paper by the same authors, we provided a theoretical foundation for the component-by-component (CBC) construction of lattice algorithms for multivariate \(L_2\) approximation in the worst case setting, for functions in a periodic space with general weight parameters. The construction led to an error bound that achieves the best possible rate of convergence for lattice algorithms. Previously available literature covered only weights of a simple form commonly known as product weights. In this paper we address the computational aspect of the construction. We develop fast CBC construction of lattice algorithms for special forms of weight parameters, including the so-called POD weights and SPOD weights which arise from PDE applications, making the lattice algorithms truly applicable in practice. With \(d\) denoting the dimension and \(n\) the number of lattice points, we show that the construction cost is \(\mathcal{O} (d\ n \log (n) + d^2 \log (d)\ n)\) for POD weights, and \(\mathcal{O}(d\ n \log (n) + d^3 \sigma^2 \ n)\) for SPOD weights of degree \(\sigma \ge 2\). The resulting lattice generating vectors can be used in other lattice-based approximation algorithms, including kernel methods or splines.Book review of: V. Gupta et al., Recent advances in constructive approximation theory.https://www.zbmath.org/1456.000272021-04-16T16:22:00+00:00"Milovanović, Gradimir V."https://www.zbmath.org/authors/?q=ai:milovanovic.gradimir-vReview of [Zbl 1400.41017].Approximation on weighted spaces with Bernstein-Chlodowsky operators.https://www.zbmath.org/1456.410032021-04-16T16:22:00+00:00"Çiçek, Harun"https://www.zbmath.org/authors/?q=ai:cicek.harunSummary: In this text, the approximation features and the speed of approximation of Modified Bernstein-Chlodowsky Operators on weighted spaces will be examined. Furthermore, definitions and some properties for weighted spaces are given. We guess the order of approximation by Voronovskaya type theorem. Lastly, some numerical examples and related figures are presented.Asymptotic greediness of the Haar system in the spaces \(L_p[0,1]\), \(1<p<\infty \).https://www.zbmath.org/1456.460162021-04-16T16:22:00+00:00"Albiac, Fernando"https://www.zbmath.org/authors/?q=ai:albiac.fernando"Ansorena, José L."https://www.zbmath.org/authors/?q=ai:ansorena.jose-luis"Berná, Pablo M."https://www.zbmath.org/authors/?q=ai:berna.pablo-manuelThe Haar system is an unconditional basis of \(L^p[0,1]\) that plays a central role in harmonic analysis and its applications. As demonstrated with particular clarity in [\textit{T. P. Hytönen}, Ann. Math. (2) 175, No. 3, 1473--1506 (2012; Zbl 1250.42036)], all Calderón-Zygmund operators (and thus a wide range of solution maps to PDE problems) can be constructed from basic operations on Haar systems. An influential result of \textit{D. L. Burkholder} [Stud. Math. 91, No. 1, 79--83 (1988; Zbl 0652.42012)] shows that, as a function of \(p\), the unconditionality constant of the Haar system behaves like \(p^{*} = \max \{p,p/(p-1)\}\). Many (optimal) quantitative results in harmonic analysis involve similar ideas (see, e.g., [Hytönen, loc.\,cit.]). Burkholder's method of proof, in particular, has been extensively developed in recent years as part of a general theory called the {Bellman function method} (see [\textit{F. Nazarov} et al., in: Systems, approximation, singular integral operators, and related topics. Proceedings of the 11th international workshop on operator theory and applications, IWOTA 2000, Bordeaux, France, June 13--16, 2000. Basel: Birkhäuser. 393--423 (2001; Zbl 0999.60064)]).
The paper under review addresses a natural question, raised by Hytönen: does the greedy constant of the Haar basis also grow linearly with \(p^{*}\) as \(p^{*}\) tends to infinity?
Recall that the greedy constant of a basis \((h_{j})_{j \in \mathbb{N}}\) is the infimum of all \(C>0\) such that
\[
\|f-\mathcal{G}_{m}(f)\| \leq C \left\|f - \sum \limits _{j \in A} a_{j}h_{j} \right\|
\]
for all \(f \in L^{p}\), all \(m \in \mathbb{N}^{*}\), all finite subsets \(A\subset \mathbb{N}\) of cardinality \(m\), and all \((a_{j})_{j \in A} \in \mathbb{R}^{m}\), where \(\mathcal{G}_{m}(f)\) denotes the \(m\)-th greedy approximation of \(f\) (i.e., the sum of \(m\) terms with the largest coefficients in the basis expansion). Qualitatively, a~basis has finite greedy constant if and only if it is unconditional and democratic (in the sense that \(\|\sum _{j \in A} h_{j}\| \leq C_{d} \|\sum _{j \in B} h_{j}\|\) for some \(C_{d}>0\) and all finite \(A,B \subset \mathbb{N}\) of equal cardinality). The question thus has two interesting quantitative components: determining how the democracy constant \(C_{d}\) depends on \(p^{*}\), and determining how the greedy constant depends on both the unconditionality constant and the democracy constant.
Known results for this second component give a linear growth in the lower bound, but only a quadratic growth in the upper bound. This is true even if one replaces democracy by super-democracy or symmetry for largest coefficients (see Section~1 of the paper). The authors nonetheless prove that the greedy constant of the Haar basis does grow linearly in \(p^{*}\). To do so, they exploit an extra property called {bi-democracy}:
\[
\left\|\sum _{j \in A} h_{j} \right\|_{p} \cdot \left\|\sum _{k \in B} h_{k} \right\|_{p'}
\leq C_{b} m , \quad |A|=|B|=m.
\]
Their proof has two parts. The first part is abstract (valid for all bi-greedy bases in Banach spaces) and shows that the greedy constant of a bi-democratic basis is bounded by a linear combination of the unconditionality constant and the bi-democratic constant. This result nicely extends previous works in the democratic case, such as [\textit{P. M. Berná} et al., Rev. Mat. Complut. 30, No. 2, 369--392 (2017; Zbl 1375.41020)]. Its proof involves a~natural separation of the indices that appear in the greedy approximation from those that do not (see Theorem~2.3).
The second part shows that the bi-democratic constant of the Haar basis is proportional to~\(p^{*}\). For the lower bound, this is proven by selecting appropriate intervals inductively (starting with \(I_{1}=[0,1)\) then picking \(I_{j+1}\) to be the left half of \(I_{j}\)). For the upper bound, the authors use two key properties of the Haar basis: the dyadic intervals form a partition of \([0,1)\), and the dual functional of the Haar functions in \(L^{p}\) are the Haar functions in \(L^{p'}\).
Reviewer: Pierre Portal (Canberra)Quasi-optimal adaptive mixed finite element methods for controlling natural norm errors.https://www.zbmath.org/1456.651642021-04-16T16:22:00+00:00"Li, Yuwen"https://www.zbmath.org/authors/?q=ai:li.yuwenSummary: For a generalized Hodge Laplace equation, we prove the quasi-optimal convergence rate of an adaptive mixed finite element method. This adaptive method can control the error in the natural mixed variational norm when the space of harmonic forms is trivial. In particular, we obtain new quasi-optimal adaptive mixed methods for the Hodge Laplace, Poisson, and Stokes equations. Comparing to existing adaptive mixed methods, the new methods control errors in both variables.Asymptotics for Hermite-Padé approximants associated with the Mittag-Leffler functions.https://www.zbmath.org/1456.300662021-04-16T16:22:00+00:00"Starovoitov, A. P."https://www.zbmath.org/authors/?q=ai:starovoitov.alexandr-p"Kechko, E. P."https://www.zbmath.org/authors/?q=ai:kechko.e-pSummary: In this article, under certain restrictions, the convergence rate of type II Hermite-Padé approximants (including nondiagonal ones) for a system \(\{{}_1F_1(1,\gamma;\lambda_jz)\}_{j=1}^k\), consisting of degenerate hypergeometric functions is found, when \(\{\lambda_j\}_{j=1}^k\) are different complex numbers, and \(\gamma\in\mathbb{C}\setminus\{0,-1,-2,\dots\}\). Without the indicated restrictions, similar statements were obtained for approximants of the indicated type, provided that the numbers \(\{\lambda_j\}_{j=1}^k\) are the roots of the equation \(\lambda^k=1\). The theorems proved in this paper complement and generalize the results obtained earlier by other authors.Universal distribution of random matrix eigenvalues near the ``birth of a cut'' transition.https://www.zbmath.org/1456.813722021-04-16T16:22:00+00:00"Eynard, B."https://www.zbmath.org/authors/?q=ai:eynard.bertrandEquichordal tight fusion frames.https://www.zbmath.org/1456.420432021-04-16T16:22:00+00:00"Mohammadpour, Mozhgan"https://www.zbmath.org/authors/?q=ai:mohammadpour.mozhgan"Kamyabi-Gol, Rajab Ali"https://www.zbmath.org/authors/?q=ai:kamyabi-gol.rajab-ali"Hodtani, Ghosheh Abed"https://www.zbmath.org/authors/?q=ai:hodtani.ghosheh-abedSummary: A Grassmannian fusion frame is an optimal configuration of subspaces of a given vector space, that are useful in some applications related to representing data in signal processing. Grassmannian fusion frames are robust against noise and erasures when the signal is reconstructed. In this paper, we present an approach to construct optimal Grassmannian fusion frames based on a given Grassmannian frame. We also analyse an algorithm for sparse fusion frames which was introduced by \textit{R. Calderbank} et al. [Adv. Comput. Math. 35, No. 1, 1--31 (2011; Zbl 1264.94042)] and present necessary and sufficient conditions for the output of that algorithm to be an optimal Grassmannian fusion frame.Trigonometric approximation by angle in classical weighted Lorentz and grand Lorentz spaces.https://www.zbmath.org/1456.420042021-04-16T16:22:00+00:00"Kokilashvili, Vakhtang"https://www.zbmath.org/authors/?q=ai:kokilashvili.vakhtang-m"Tsanava, Tsira"https://www.zbmath.org/authors/?q=ai:tsanava.tsiraA survey of the main results on angular approximations of functions in Lebesgue spaces is given in [\textit{M. K. Potapov} et al., Surv. Approx. Theory 8, 1--57 (2013; Zbl 1285.26008)]. For the \(L_p\)-spaces with Muckenhoupt weights, some of these results were extended in [\textit{R. Akgün}, Complex Var. Elliptic Equ. 64, No. 2, 330--351 (2019; Zbl 1405.42002)]. As further generalizations, the authors formulate several theorems for the weighted Lorentz spaces and for the grand Lorentz spaces. It is noted that the proofs of the presented results will be published in the Georgian Mathematical Journal.
Reviewer: Yuri A. Farkov (Moskva)Approximation by trigonometric polynomials in the framework of weighted fully measurable grand Lorentz spaces.https://www.zbmath.org/1456.420032021-04-16T16:22:00+00:00"Kokilashvili, Vakhtang"https://www.zbmath.org/authors/?q=ai:kokilashvili.vakhtang-mSummary: In this note we present the fundamental Bernstein and Nikol'skii type inequalities in weighted fully measurable grand Lorentz spaces. These inequalities we apply to obtain the direct and inverse approximation theorems in approximable subspaces of aforementioned function spaces.On \(L_p\)-error of bivariate polynomial interpolation on the square.https://www.zbmath.org/1456.410012021-04-16T16:22:00+00:00"Kolomoitsev, Yurii"https://www.zbmath.org/authors/?q=ai:kolomoitsev.yurii-s"Lomako, Tetiana"https://www.zbmath.org/authors/?q=ai:lomako.tetiana"Prestin, Jürgen"https://www.zbmath.org/authors/?q=ai:prestin.jurgenSummary: We obtain estimates of the \(L_p\)-error of the bivariate polynomial interpolation on the Lissajous-Chebyshev node points for wide classes of functions including non-smooth functions of bounded variation in the sense of Hardy-Krause. The results show that \(L_p\)-errors of polynomial interpolation on the Lissajous-Chebyshev nodes have almost the same behavior as the polynomial interpolation in the case of the tensor product Chebyshev grid.Approximating smooth functions by deep neural networks with sigmoid activation function.https://www.zbmath.org/1456.410052021-04-16T16:22:00+00:00"Langer, Sophie"https://www.zbmath.org/authors/?q=ai:langer.sophieSummary: We study the power of deep neural networks (DNNs) with sigmoid activation function. Recently, it was shown that DNNs approximate any \(d\)-dimensional, smooth function on a compact set with a rate of order \(W^{-p\slash d}\), where \(W\) is the number of nonzero weights in the network and \(p\) is the smoothness of the function. Unfortunately, these rates only hold for a special class of sparsely connected DNNs. We ask ourselves if we can show the same approximation rate for a simpler and more general class, i.e., DNNs which are only defined by its width and depth. In this article we show that DNNs with fixed depth and a width of order \(M^d\) achieve an approximation rate of \(M^{- 2 p}\). As a conclusion we quantitatively characterize the approximation power of DNNs in terms of the overall weights \(W_0\) in the network and show an approximation rate of \(W_0^{- p \slash d}\). This more general result finally helps us to understand which network topology guarantees a special target accuracy.Estimates of the approximations by Zygmund sums in Morrey-Smirnov classes of analytic functions.https://www.zbmath.org/1456.300642021-04-16T16:22:00+00:00"Jafarov, S. Z."https://www.zbmath.org/authors/?q=ai:jafarov.sadulla-zSummary: In the present work, we investigate the approximation of the functions by Zygmund means of Fourier series in Morrey spaces \(L^{p, \lambda}(\mathbb{T})\), \(0< \lambda \leq 2 \), \(1 < p < \infty\) in the terms of the modulus of smoothness. The obtained results are applied to estimate the approximation of functions by Zygmund sums of Faber series in Morrey-Smirnov classes defined on simply connected domains of the complex plane. In this case, the obtained estimation depends on the sequence of the best approximation in Morrey-Smirnov classes.Angular trigonometric approximation in the framework of new scale of function spaces.https://www.zbmath.org/1456.420052021-04-16T16:22:00+00:00"Kokilashvili, V."https://www.zbmath.org/authors/?q=ai:kokilashvili.vakhtang-m"Tsanava, Ts."https://www.zbmath.org/authors/?q=ai:tsanava.tsiraSummary: In this note we announce our results on approximation by angle of functions of two variables in weighted Lebesgue spaces with mixed norms. The related problems of multidimensional Fourier Analysis are explored as well.Supercongruences arising from hypergeometric series identities.https://www.zbmath.org/1456.110052021-04-16T16:22:00+00:00"Liu, Ji-Cai"https://www.zbmath.org/authors/?q=ai:liu.jicaiThis paper refines a supercongruence of \textit{T. Kilbourn} [Acta Arith. 123, No. 4, 335--348 (2006; Zbl 1170.11008)] about the identity \[ a(p)=p^3-2p^2-7-N(p) \] studied by \textit{S. Ahlgren} and \textit{K. Ono} [J. Reine Angew. Math. 518, 187--212 (2000; Zbl 0940.33002)], by \textit{B. van Geemen} and \textit{N. O. Nygaard} [J. Number Theory 53, No. 1, 45--87 (1995; Zbl 0838.11047)], and by \textit{H. A. Verrill} [CRM Proc. Lecture Notes 19, 333--340. Providence, RI: Amer. Math. Soc. (1999; Zbl 0942.14022] in connection to the modular Calabi-Yau threefold for odd primes \(p\) \[ x+\frac{1}{x}+y+\frac{1}{y}+z+\frac{1}{z}+w+\frac{1}{w}=0 \] associated with truncated hypergeometric series.
Namely, the author establishes that \[a(p) \equiv p \cdot {{_{4}F_3} \left[ \begin{matrix} \frac{1}{2}, & \frac{1}{2}, & \frac{1}{2}, & \frac{1}{2}\;\\ & 1, & \frac{3}{4}, & \frac{5}{4} \end{matrix} \Big| \; 1 \right]}_ \frac{ p-1}{2} \pmod {p^3}\] for any prime \(p \geq 5 \).
In addition, the paper gives a ``human proof'' of a supercongruence already found by the author [J. Math. Anal. Appl. 471, 613--622 (2019; Zbl 1423.11015)], via the Mathematica package \(Sigma\) supplied by \textit{C. Schneider} [Sémin. Lothar. Comb. 56, B56b, 36 p. (2006; Zbl 1188.05001)], as extension of the \(p\)-adic analogue of a Ramanujan's identity conjectured by \textit{L. van Hamme} [Lect. Notes Pure Appl. Math. 192, 223--236 (1997; Zbl 0895.11051)].
Beyond basic properties of the gamma function (both classical and \(p\)-adic), the Taylor's expansion, and the Wolstenholme's congruence, the theorem-proving recalls some results from \textit{W. N. Bailey} [Generalized hypergeometric series. London: Cambridge University Press (1935; Zbl 0011.02303)], from \textit{L. Long} and \textit{R. Ramakrishna} [Adv. Math. 290, 773--808 (2016; Zbl 1336.33018)], and from \textit{F. J. W. Whipple} [Proc. London Math Soc. 24, 247--263 (1926; JFM 51.0283.03)].
Reviewer: Enzo Bonacci (Latina)Direct and converse theorems for King operators.https://www.zbmath.org/1456.410072021-04-16T16:22:00+00:00"Finta, Zoltán"https://www.zbmath.org/authors/?q=ai:finta.zoltanAuthor's abstract: For the sequence of King operators, we establish a direct approximation theorem via the first order Ditzian-Totik modulus of smoothness, and a converse approximation theorem of Berens-Lorentz-type.
Reviewer: Sorin Gheorghe Gal (Oradea)Further improvement on bounds for \(L\)-functions related to \(\mathrm{GL}(3)\).https://www.zbmath.org/1456.110852021-04-16T16:22:00+00:00"Sun, Haiwei"https://www.zbmath.org/authors/?q=ai:sun.haiwei"Ye, Yangbo"https://www.zbmath.org/authors/?q=ai:ye.yangboModelling of the axisymmetric precision electrochemical shaping.https://www.zbmath.org/1456.300742021-04-16T16:22:00+00:00"Zhitnikov, Vladimir Pavlovich"https://www.zbmath.org/authors/?q=ai:zhitnikov.vladimir-pavlovich"Sherykhalina, Nataliya Mikhaĭlovna"https://www.zbmath.org/authors/?q=ai:sherykhalina.nataliya-mikhailovna"Porechnyĭ, Sergeĭ Sergeevich"https://www.zbmath.org/authors/?q=ai:porechnyi.sergei-sergeevich"Sokolova, Aleksandra Alekseevna"https://www.zbmath.org/authors/?q=ai:sokolova.aleksandra-alekseevnaSummary: The problem on modelling of a precision shaping and boundary conditions are formulated according to Faraday's law and with applying of stepwise dependence current efficiency on current density. The problem is reduced to the solution of a boundary problem for definition of two analytical functions of the complex variable. The first function is a conformal mapping of region of parametrical variable on the physical plane. In order to determine this function we use the Schwartz's integral and a spline interpolation. Unlike a plane problem for determination of potential and stream function of an axisymmetric field, the integration transformations of the second analytical function are used. The analytical function is defined in the form of a sum of two addends. The first addend takes into account the singularities of the function so that the second addend has no singularities. The second function is defined by the Schwartz's integral. Interpolation by spline functions is carried out, where the spline coefficients are derivatives of these functions by means of which the intensity vector components are calculated. We propose the method to solve the axisymmetric stationary problems, which differs from the known methods by the accuracy. By means of the method, we obtain the numerical results, describing the workpiece form. The error estimation of the obtained results is carried out. Also, we show qualitative coincidence with results of plane problem solution.Topics in uniform approximation of continuous functions.https://www.zbmath.org/1456.410092021-04-16T16:22:00+00:00"Bucur, Ileana"https://www.zbmath.org/authors/?q=ai:bucur.ileana-gabriela"Paltineanu, Gavriil"https://www.zbmath.org/authors/?q=ai:paltineanu.gavriilThis book is not only a nice survey on approximation results of continuous functions but it also contains many results of the authors.
It presents some of the most important results on the topic, from the approximation of real functions defined on compact intervals to abstract
approximation theory in locally convex lattices of \((M)\) type. The contents of the book can briefly be described as follows.
Chapter 1 contains in Section 1.1 the classical Weierstrass' theorems, by using in the algebraic case the Khun's idea of proof. Section 1.2 presents the
classical Korovkin's theorem on approximation by positive and linear operators and the classical proof of Weierstrass' theorem by using the Bernstein's polynomials. In Section 1.3, by using the idea of Brosowski-Deutsch the classical generalization of Weierstrass' theorem known as Stone-Weierstrass theorem is proved. Section 1.4 contains an elegant proof due to T. J. Ransford, which uses the Bernoulli's inequality, to obtain a generalization of the Stone-Weierstrass theorem to non-self adjoint algebras originally given by E. Bishop. Section 1.5 presents the results of the authors concerning some generalizations of Stone-Weierstrass' theorems and of Prolla's result, concerning the subsets of \(C(X; [0, 1])\) with the so-called von Neumann property.
Chapter 2 treats various approximation theorems for continuous functions defined on a locally compact space. The natural frame for such a kind of theorems are the so-called weighted spaces, introduced by Leopold Nachbin. The main results in Sections 2.3, 2.4 and 2.5 deal with Stone-Weierstrass-type approximation results for a vector subspace or a convex sub-cone of a weighted space and with generalizations to vector functions.
In Chapter 3 the classical Bernstein's and Nachbin's results on approximation of continuously differentiable functions
are presented.
Chapter 4 is the core of originality, being entirely based on the papers [14, 15-21, 28-30]. In essence, it is devoted to the generalizations of the results in the previous chapters, to the abstract case of locally convex lattices of type \((M)\), abstract spaces which generalize the weighted spaces \(CV_{0}(X, \mathbb{R})\) and \(CV_{0}(X, \mathbb{C})\). Defining the notions of antisymmetric ideals in real and
complex locally convex lattices of type \((M)\),
theorems of approximation of the elements of such a lattice by elements belonging to some subspaces are proved.
This is an interesting book for researchers working in the general topic of the theory of functions and in approximation theory, in particular. It can also be very useful for courses addressed to graduate and Ph.d students in mathematics.
Reviewer: Sorin Gheorghe Gal (Oradea)Banded operational matrices for Bernstein polynomials and application to the fractional advection-dispersion equation.https://www.zbmath.org/1456.352142021-04-16T16:22:00+00:00"Jani, M."https://www.zbmath.org/authors/?q=ai:jani.mostafa|jani.mahendra"Babolian, E."https://www.zbmath.org/authors/?q=ai:babolian.esmail|babolian.esmaeil"Javadi, S."https://www.zbmath.org/authors/?q=ai:javadi.shahnam|javadi.samaneh|javadi.seyed-mohammad-mahdi|javadi.sonya|javadi.samanech"Bhatta, D."https://www.zbmath.org/authors/?q=ai:bhatta.dambaru-d|bhatta.dilliSummary: In the papers, dealing with derivation and applications of operational matrices of Bernstein polynomials, a basis transformation, commonly a transformation to power basis, is used. The main disadvantage of this method is that the transformation may be ill-conditioned. Moreover, when applied to the numerical simulation of a functional differential equation, it leads to dense operational matrices and so a dense coefficient matrix is obtained. In this paper, we present a new property for Bernstein polynomials. Using this property, we build exact banded operational matrices for derivatives of Bernstein polynomials. Next, as an application, we propose a new numerical method based on a Petrov-Galerkin variational formulation and the new operational matrices utilizing the dual Bernstein basis for the time-fractional advection-dispersion equation. We show that the proposed method leads to a narrow-banded linear system and so less computational effort is required to obtain the desired accuracy for the approximate solution. We also obtain the error estimation for the method. Some numerical examples are provided to demonstrate the efficiency of the method and to support the theoretical claims.Generalized weighted statistical convergence of double sequences and applications.https://www.zbmath.org/1456.400032021-04-16T16:22:00+00:00"Cinar, Muhammed"https://www.zbmath.org/authors/?q=ai:cinar.muhammed"Et, Mikail"https://www.zbmath.org/authors/?q=ai:et.mikailSummary: In this paper we introduce the concept generalized weighted statistical convergence of double sequences. Some relations between weighted $(\lambda,\mu)$-statistical convergence and strong $(\overline N_{\lambda,\mu},p,q,\alpha,\beta)$-summablity of double sequences are examined. Furthermore we apply our new summability method to prove a Korovkin type theorem.