Recent zbMATH articles in MSC 41Ahttps://www.zbmath.org/atom/cc/41A2022-05-16T20:40:13.078697ZWerkzeugA probability estimate for the discrepancy of Korobov lattice pointshttps://www.zbmath.org/1483.111562022-05-16T20:40:13.078697Z"Illarionov, Andrei A."https://www.zbmath.org/authors/?q=ai:illarionov.andrei-aThe discrete case of the mixed joint universality for a class of certain partial zeta-functionshttps://www.zbmath.org/1483.111912022-05-16T20:40:13.078697Z"Kačinskaitė, Roma"https://www.zbmath.org/authors/?q=ai:kacinskaite.roma"Matsumoto, Kohji"https://www.zbmath.org/authors/?q=ai:matsumoto.kohjiAuthors' abstract: We give a new type of mixed discrete joint universality properties, which is satisfied by a wide class of zeta-functions. We study the universality for a certain modification of Matsumoto zeta-functions \(\varphi_h(s)\) and a collection of periodic Hurwitz zeta-functions \(\zeta (s;\alpha;\mathfrak B)\) under the condition that the common difference of arithmetical progression \(h > 0\) is such that \(\exp \{ \frac{2\pi}h\}\) is a rational number and parameter \(\alpha\) is a transcendental number.
Reviewer: Anatoly N. Kochubei (Kyïv)Joint discrete universality for periodic zeta-functions. IIIhttps://www.zbmath.org/1483.111972022-05-16T20:40:13.078697Z"Laurinčikas, Antanas"https://www.zbmath.org/authors/?q=ai:laurincikas.antanasSummary: In the paper, a joint theorem on the approximation of collections of analytic functions by generalized discrete shifts of zeta-functions with periodic coefficients is obtained. The latter result extend theorems of [Part I, Zbl 1441.11235].
For Part I and II, see [ibid. 42, No. 5, 687--699 (2019; Zbl 1441.11235); ibid. 43, No. 12, 1765--1779 (2020; Zbl 1457.11125)].Kolmogorov inequalities for norms of Marchaud-type fractional derivatives of multivariate functionshttps://www.zbmath.org/1483.260192022-05-16T20:40:13.078697Z"Parfinovych, N. V."https://www.zbmath.org/authors/?q=ai:parfinovych.nataliia-viktorivna"Pylypenko, V. V."https://www.zbmath.org/authors/?q=ai:pylypenko.v-vSummary: We obtain new sharp Kolmogorov type inequalities, estimating the norm of mixed Marchaud type derivative of multivariate function through the \(C\)-norm of function itself and its norms in Hölder spaces.Asymptotics of compound meanshttps://www.zbmath.org/1483.260262022-05-16T20:40:13.078697Z"Hilberdink, Titus"https://www.zbmath.org/authors/?q=ai:hilberdink.titus-wIn this paper, the author studies bivariate means \(m\) and \(M\) which can be formed from sequences \(a_n, b_n\) defined recursively by \(a_{n+1} = m(a_n, b_n)\), \(b_{n+1} = M(a_n, b_n)\) with \(a_0, b_0 > 0\). He investigates under mild conditions when these means will converge to a new mean \(\mathcal{M}(a_0,b_0)\), called a compound mean. The special case \(m\) and \(M\) are homogeneous, it is shown that \(\mathcal{M}\) is also homogeneous and satisfies a functional equation. Further, the author studies the asymptotic behaviour of \(\mathcal{M}(1, x)\) as \(x \to \infty\) given that of \(m\) and \(M\), and obtains the main term up to a possible oscillatory function. The oscillatory behaviour is also investigated if \(m\) and \(M\) are coming from some well-known classes of means. Some numerical computations are also reported which show that the oscillations are generic.
Reviewer: James Adedayo Oguntuase (Abeokuta)Weighted Chebyshev polynomials on compact subsets of the complex planehttps://www.zbmath.org/1483.300232022-05-16T20:40:13.078697Z"Novello, Galen"https://www.zbmath.org/authors/?q=ai:novello.galen"Schiefermayr, Klaus"https://www.zbmath.org/authors/?q=ai:schiefermayr.klaus"Zinchenko, Maxim"https://www.zbmath.org/authors/?q=ai:zinchenko.maximSummary: We study weighted Chebyshev polynomials on compact subsets of the complex plane with respect to a bounded weight function. We establish existence and uniqueness of weighted Chebyshev polynomials and derive weighted analogs of Kolmogorov's criterion, the alternation theorem, and a characterization due to Rivlin and Shapiro. We derive invariance of the Widom factors of weighted Chebyshev polynomials under polynomial pre-images and a comparison result for the norms of Chebyshev polynomials corresponding to different weights. Finally, we obtain a lower bound for the Widom factors in terms of the Szegő integral of the weight function and discuss its sharpness.
For the entire collection see [Zbl 1479.47003].Bounded extremal problems in Bergman and Bergman-Vekua spaceshttps://www.zbmath.org/1483.300892022-05-16T20:40:13.078697Z"Delgado, Briceyda B."https://www.zbmath.org/authors/?q=ai:delgado.briceyda-b"Leblond, Juliette"https://www.zbmath.org/authors/?q=ai:leblond.julietteSummary: We analyze Bergman spaces \(A_f^p(\mathbb{D})\) of generalized analytic functions of solutions to the Vekua equation \(\bar{\partial}w = (\bar{\partial}f/f)\bar{w}\) in the unit disc of the complex plane, for Lipschitz-smooth non-vanishing real valued functions \(f\) and \(1<p<\infty\). We consider a family of bounded extremal problems (best constrained approximation) in the Bergman space \(A^p(\mathbb{D})\) and in its generalized version \(A^p_f(\mathbb{D})\), that consists in approximating a function in subsets of \(\mathbb{D}\) by the restriction of a function belonging to \(A^p(\mathbb{D})\) or \(A^p_f(\mathbb{D})\) subject to a norm constraint. Preliminary constructive results are provided for \(p = 2\).Bernstein-Markov property for compact sets in \(\mathbb{C}^d\)https://www.zbmath.org/1483.320012022-05-16T20:40:13.078697Z"Hoang Thieu Anh"https://www.zbmath.org/authors/?q=ai:hoang-thieu-anh."Nguyen Quang Dieu"https://www.zbmath.org/authors/?q=ai:nguyen-quang-dieu."Tang Van Long"https://www.zbmath.org/authors/?q=ai:tang-van-long.Summary: Given a compact set \(K\) in \(\mathbb{C}^d\). We concern with the Bernstein-Markov property of the pair \((K,\mu )\) where \(\mu\) is a finite positive Borel measure with compact support \(K\). In particular, we are able to give a class of \((K,\mu )\) having the Bernstein-Markov property with the measure \(\mu\) satisfies a rather weak density condition. Using this result, we construct a pair \((K,\mu )\) satisfying the Bernstein-Markov property which is not covered by the known results in [\textit{T. Bloom}, Indiana Univ. Math. J. 46, No. 2, 427--452 (1997; Zbl 0930.42013); \textit{T. Bloom} and \textit{N. Levenberg}, Trans. Am. Math. Soc. 351, No. 12, 4753--4767 (1999; Zbl 0933.31007)]. Another main result of the note is a \textit{weak} characterization of Bernstein-Markov property in terms of Chebyshev constants.Faedo-Galerkin approximation of mild solutions of fractional functional differential equationshttps://www.zbmath.org/1483.341072022-05-16T20:40:13.078697Z"Vanterler da Costa Sousa, José"https://www.zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.jose"Fečkan, Michal"https://www.zbmath.org/authors/?q=ai:feckan.michal"de Oliveira, Edmundo Capelas"https://www.zbmath.org/authors/?q=ai:de-oliveira.edmundo-capelasSummary: In the paper, we discuss the existence and uniqueness of mild solutions of a class of fractional functional differential equations in Hilbert space separable using the Banach fixed point theorem technique. In this sense, Faedo-Galerkin approximation to the solution is studied and demonstrated some convergence results.Design of a mode converter using thin resonant ligamentshttps://www.zbmath.org/1483.350142022-05-16T20:40:13.078697Z"Chesnel, Lucas"https://www.zbmath.org/authors/?q=ai:chesnel.lucas"Heleine, Jérémy"https://www.zbmath.org/authors/?q=ai:heleine.jeremy"Nazarov, Sergei A."https://www.zbmath.org/authors/?q=ai:nazarov.sergei-aleksandrovichSummary: The goal of this work is to design an acoustic mode converter. The wave number is fixed so that two modes can propagate. We explain how to construct geometries such that the energy of the modes is completely transmitted and additionally the mode 1 is converted into the mode 2 and conversely. To proceed, we work in a symmetric waveguide made of two branches connected by two thin ligaments whose lengths and positions are carefully tuned. The approach is based on asymptotic analysis for thin ligaments around resonance lengths. We also provide numerical results to illustrate the theory.Long-time asymptotics for the focusing Fokas-Lenells equation in the solitonic region of space-timehttps://www.zbmath.org/1483.351752022-05-16T20:40:13.078697Z"Cheng, Qiaoyuan"https://www.zbmath.org/authors/?q=ai:cheng.qiaoyuan"Fan, Engui"https://www.zbmath.org/authors/?q=ai:fan.enguiSummary: We study the long-time asymptotic behavior of the focusing Fokas-Lenells (FL) equation
\[ u_{x t} + \alpha \beta^2 u - 2 i \alpha \beta u_x - \alpha u_{x x} - i \alpha \beta^2 | u |^2 u_x = 0\]
with generic initial data in a Sobolev space which supports bright soliton solutions. The FL equation is an integrable generalization of the well-known Schrodinger equation, and also linked to the derivative Schrodinger model, but it exhibits several different characteristics from theirs. (i) The Lax pair of the FL equation involves an additional spectral singularity at \(k = 0\). (ii) Four stationary phase points will appear during asymptotic analysis, which require a more detailed necessary description to obtain the long-time asymptotics of the focusing FL equation. Based on the Riemann-Hilbert problem for the initial value problem of the focusing FL equation, we show that inside any fixed time-spatial cone
\[ \mathcal{C} ( x_1 , x_2 , v_1 , v_2 ) = \{ ( x , t ) \in \mathbb{R}^2 | x = x_0 + v t , x_0 \in [ x_1 , x_2 ] , v \in [ v_1 , v_2 ] \},\]
the long-time asymptotic behavior of the solution \(u(x, t)\) for the focusing FL equation can be characterized with an \(N(\mathcal{I})\)-soliton on discrete spectrums and a leading order term \(\mathcal{O}( t^{- 1 / 2})\) on continuous spectrum up to a residual error order \(\mathcal{O}( t^{- 3 / 4})\). The main tool is a \(\overline{\partial} \)-generalization of the Deift-Zhou nonlinear steepest descent method.On Shallit's minimization problemhttps://www.zbmath.org/1483.371132022-05-16T20:40:13.078697Z"Sadov, S. Yu."https://www.zbmath.org/authors/?q=ai:sadov.sergey-yuSummary: In Shallit's problem [\textit{PJ. Shallit}, SIAM Rev., 36, No. 3, 490--491 (1994)], it was proposed to justify a two-term asymptotics of the minimum of a rational function of \(n\) variables defined as the sum of a special form whose number of terms is of order \(n^2\) as \(n\to\infty \). Of particular interest is the second term of this asymptotics (``Shallit's constant''). The solution published in SIAM Review presented an iteration algorithm for calculating this constant, which contained some auxiliary sequences with certain properties of monotonicity. However, a rigorous justification of the properties, necessary to assert the convergence of the iteration process, was replaced by a reference to numerical data. In the present paper, the gaps in the proof are filled on the basis of an analysis of the trajectories of a two-dimensional dynamical system with discrete time corresponding to the minimum points of \(n\)-sums. In addition, a sharp exponential estimate of the remainder in Shallit's asymptotic formula is obtained.Box-counting dimension and analytic properties of hidden variable fractal interpolation functions with function contractivity factorshttps://www.zbmath.org/1483.410012022-05-16T20:40:13.078697Z"Yun, CholHui"https://www.zbmath.org/authors/?q=ai:yun.cholhui"Ri, MiGyong"https://www.zbmath.org/authors/?q=ai:ri.mi-gyongSummary: We estimate the bounds for box-counting dimension of hidden variable fractal interpolation functions (HVFIFs) and hidden variable bivariate fractal interpolation functions (HVBFIFs) with four function contractivity factors and present analytic properties of HVFIFs which are constructed to ensure more flexibility and diversity in modeling natural phenomena. Firstly, we construct the HVFIFs and analyze their smoothness and stability. Secondly, we obtain the lower and upper bounds for box-counting dimension of the HVFIFs. Finally, in the similar way, we get the lower and upper bounds for box-counting dimension of HVBFIFs in [\textit{C.-H. Yun} and \textit{M.-K. Ri}, Asian-Eur. J. Math. 12, No. 2, Article ID 1950021, 15 p. (2019; Zbl 1409.28003)].Extremal problems for non-periodic splines on real domain and their derivativeshttps://www.zbmath.org/1483.410022022-05-16T20:40:13.078697Z"Danchenko, K. A."https://www.zbmath.org/authors/?q=ai:danchenko.k-a"Kofanov, V. A."https://www.zbmath.org/authors/?q=ai:kofanov.v-aSummary: We consider the Bojanov-Naidenov problem over the set \( \sigma_{h,r}\) of all non-periodic splines \(s\) of order \(r\) and minimal defect with knots at the points \(kh, k \in \mathbb{Z} \). More exactly, for given \(n, r \in \mathbb{N}\); \(p, A > 0\) and any fixed interval \([a, b] \subset \mathbb{R} \) we solve the following extremal problem
\[ \int\limits_a^b |x(t)|^q dt \rightarrow \sup,\quad q \geqslant p,\tag{1}\]
over the classes
\[ \sigma_{h,r}^p(A) := \left\{ s(\cdot + \tau) \colon s \in \sigma_{h,r}, \| s \|_{p, \delta} \leqslant A \| \varphi_{\lambda, r} \|_{p, \delta}, \quad \delta \in (0, h], \quad \tau \in \mathbb{R} \right\},\]
where
\[ \| x \|_{p, \delta} := \sup \left\{ \| x \|_{L_p[a,b]} \colon a, b \in \mathbb{R}, \quad 0 < b - a \leqslant \delta \right\},\]
and \( \varphi_{\lambda, r} \) is a \((2\pi / \lambda)\)-periodic spline of Euler of order \(r\).
In particularly, for \(k = 1, \dots, r - 1\) we solve the extremal problem
\[ \int\limits_a^b |x^{(k)}(t)|^q dt \rightarrow \sup,\quad q \geqslant 1, \tag{2}\]
over the classes \( \sigma_{h,r}^p (A)\).
Note that the problems (1) and (2) were solved earlier on the classes
\[ \sigma_{h,r}(A, p) := \left\{ s(\cdot + \tau) \colon s \in \sigma_{h,r}, L(s)_p \leqslant AL(\varphi_{n,r})_p, \quad \tau \in \mathbb{R} \right\},\]
where
\[ L(x)_p := \sup \left\{ \| x \|_{L_p[a, b]} \colon a, b \in \mathbb{R}, |x(t)| > 0, t \in (a, b) \right\}.\]
We prove that the classes \( \sigma_{h,r}^p (A)\) are wider than the classes \( \sigma_{h,r}(A,p)\). Similarly, we solve the analog of Erdös problem about the characterization of the spline \(s \in \sigma_{h,r}^p(A)\) that has maximal arc length over a fixed interval \([a, b] \subset \mathbb{R} \).The Bojanov-Naidenov problem for trigonometric polynomials and periodic splineshttps://www.zbmath.org/1483.410032022-05-16T20:40:13.078697Z"Asadova, E. V."https://www.zbmath.org/authors/?q=ai:asadova.e-v"Kofanov, V. A."https://www.zbmath.org/authors/?q=ai:kofanov.v-aSummary: For given \(n, r \in \mathbb{N}\); \(p, A > 0\) and any fixed interval \([a,b] \subset \mathbb{R} \) we solve the extremal problem \( \int\limits_a^b |x(t)|^q dt \rightarrow \sup\), \(q \geqslant p\), over sets of trigonometric polynomials \(T\) of order \( \leqslant n\) and \(2\pi \)-periodic splines \(s\) of order \(r\) and minimal defect with knots at the points \(k\pi / n\), \(k \in \mathbb{Z} \), such that \(\| T \|_{p, \delta} \leqslant A \| \sin n (\cdot) \|_{p, \delta} \leqslant A \| \varphi_{n,r} \|_{p, \delta}\), \(\delta \in (0, \pi / n]\), where \(\| x \|_{p, \delta} := \sup \{ \| x \|_{L_p[a,b]} \colon a, b \in \mathbb{R}, 0 < b - a < \delta\} \) and \( \varphi_{n, r} \) is the \((2\pi / n)\)-periodic spline of Euler of order \(r\). In particular, we solve the same problem for the intermediate derivatives \(x^{(k)}\), \(k = 1, \dots, r-1\), with \(q \geqslant 1\).Multipoint Padé approximation of the psi functionhttps://www.zbmath.org/1483.410042022-05-16T20:40:13.078697Z"Sorokin, V. N."https://www.zbmath.org/authors/?q=ai:sorokin.vladimir-nSummary: The Newton interpretation of the psi function by rational functions with free poles is studied. A discrete formula for the Rodrigues determinants is obtained and the limit distribution of their zeros is found. The corresponding equilibrium problem of the theory of logarithmic potential is obtained.Voronovskaya type results for special sequences of operatorshttps://www.zbmath.org/1483.410052022-05-16T20:40:13.078697Z"Acu, Ana-Maria"https://www.zbmath.org/authors/?q=ai:acu.ana-maria"Dancs, Madalina"https://www.zbmath.org/authors/?q=ai:dancs.madalina"Heilmann, Margareta"https://www.zbmath.org/authors/?q=ai:heilmann.margareta"Paşca, Vlad"https://www.zbmath.org/authors/?q=ai:pasca.vlad"Rasa, Ioan"https://www.zbmath.org/authors/?q=ai:rasa.ioanHere the authors present Voronovskaya type formulas that can be ``differentiated'' and which are associated with operators preserving two prescribed functions. Then the Bernstein-Schnabl type operators \(L_n\) are considered with certain new properties. In particular, its Voronovskaya formula can be ``differentiated'' and each \(L_n\) is invariant under the Kantorovich type modification. The moments of each operator \(L_n\) form a sequence of Appell polynomials, while the central moments are constant functions.
Reviewer: Vijay Gupta (New Delhi)A note on Kantorovich type Bernstein Chlodovsky operators which preserve exponential functionhttps://www.zbmath.org/1483.410062022-05-16T20:40:13.078697Z"Aral, Ali"https://www.zbmath.org/authors/?q=ai:aral.ali"Ari, Didem Aydin"https://www.zbmath.org/authors/?q=ai:ari.didem-aydin"Yılmaz, Başar"https://www.zbmath.org/authors/?q=ai:yilmaz.basarThe authors consider an integral extension of Bernstein-Chlodovsky operators, which preserve the exponential function. Inspired by the Bernstein-Chlodovsky operators they defined these operators' integral extension using a different technique. They use weighted approximation properties, including a weighted uniform convergence and a weighted quantitative theorem in terms of an exponentially weighted modulus of continuity. Furthermore, they also prove a Voronovskaya-type theorem.
Reviewer: Naokant Deo (Delhi)Connections between the approximation orders of positive linear operators and their max-product counterpartshttps://www.zbmath.org/1483.410072022-05-16T20:40:13.078697Z"Coroianu, Lucian"https://www.zbmath.org/authors/?q=ai:coroianu.lucian-c"Costarelli, Danilo"https://www.zbmath.org/authors/?q=ai:costarelli.danilo"Gal, Sorin G."https://www.zbmath.org/authors/?q=ai:gal.sorin-gheorghe"Vinti, Gianluca"https://www.zbmath.org/authors/?q=ai:vinti.gianlucaAuthors establish some direct connections between arbitrary positive linear operators and their corresponding nonlinear (more exactly sublinear) max-product versions, with respect to uniform and \(L^p\) convergence. There are numerous concrete examples of approximation operators, such as Bernstein-type operators, neural network operators, sampling operators and others, where the linear and the max-product versions converge both uniformly. Here, from the quantitative uniform approximation result for an arbitrary sequence of positive linear operators, a simple general method, a quantitative uniform approximation result for its max-product counterpart are deduced. Also convergence with respect to the \(L^p\)-norm involving the well-known K-functionals, when the supremum of the kernel is bounded from below is established.
Reviewer: Vijay Gupta (New Delhi)Converse theorem of the approximation theory of functions in Morrey-Smirnov classes related to the derivatives of functionshttps://www.zbmath.org/1483.410082022-05-16T20:40:13.078697Z"Jafarov, Sadulla Z."https://www.zbmath.org/authors/?q=ai:jafarov.sadulla-zSummary: Let \(G \subset C\) be a finite Jordan domain with a Dini-smooth boundary. In this study an inverse theorem for polynomial approximation in the Morrey-Smirnov spaces \(E^{p,\lambda}(G)\), \(0< \lambda \leq 1\) and \(1< p <\infty \), is proved.Reconstruction of two approximation processes in order to reproduce \(e^{ax}\) and \(e^{2ax}\), \(a>0\)https://www.zbmath.org/1483.410092022-05-16T20:40:13.078697Z"Yılmaz, Başar"https://www.zbmath.org/authors/?q=ai:yilmaz.basar"Uysal, Gümrah"https://www.zbmath.org/authors/?q=ai:uysal.gumrah"Aral, Ali"https://www.zbmath.org/authors/?q=ai:aral.aliThe authors proposed two modifications for Gauss-Weierstrass operators and moment-type operators which fix \(e^{ax}\) and \(e^{2ax}\), \(a>0\). They studied weighted approximation and proved Voronovskaya-type theorems in exponentially weighted spaces. Using the modulus of continuity in exponentially weighted spaces, they obtained some global smoothness preservation properties, and they gave a comparison result for Gauss-Weierstrass operators. Finally, they provided some graphical representations.
Reviewer: Naokant Deo (Delhi)Complex best \(r\)-term approximations almost always exist in finite dimensionshttps://www.zbmath.org/1483.410102022-05-16T20:40:13.078697Z"Qi, Yang"https://www.zbmath.org/authors/?q=ai:qi.yang"Michałek, Mateusz"https://www.zbmath.org/authors/?q=ai:michalek.mateusz"Lim, Lek-Heng"https://www.zbmath.org/authors/?q=ai:lim.lek-hengSummary: We show that in finite-dimensional nonlinear approximations, the best \(r\)-term approximant of a function \(f\) almost always exists over \(\mathbb{C}\) but that the same is not true over \(\mathbb{R}\), i.e., the infimum \(\inf_{f_1, \dots, f_r \in D} \| f - f_1 - \dots - f_r \|\) is almost always attainable by complex-valued functions \(f_1, \ldots, f_r\) in \(D\), a set (dictionary) of functions (atoms) with some desired structures. Our result extends to functions that possess properties like symmetry or skew-symmetry under permutations of arguments. When \(D\) is the set of separable functions, this is the best rank-\(r\) tensor approximation problem. We show that over \(\mathbb{C}\), any tensor almost always has a unique best rank-\(r\) approximation. This extends to other notions of ranks such as symmetric and alternating ranks, to best \(r\)-block-terms approximations, and to best approximations by tensor networks. Applied to sparse-plus-low-rank approximations, we obtain that for any given \(r\) and \(k\), a general tensor has a unique best approximation by a sum of a rank-\(r\) tensor and a \(k\)-sparse tensor with a fixed sparsity pattern; a problem arising in covariance estimation of Gaussian model with \(k\) observed variables conditionally independent given \(r\) hidden variables. The existential (but not uniqueness) part of our result also applies to best approximations by a sum of a rank-\(r\) tensor and a \(k\)-sparse tensor with no fixed sparsity pattern, and to tensor completion problems.Asymptotic relations for the products of elements of some positive sequenceshttps://www.zbmath.org/1483.410112022-05-16T20:40:13.078697Z"Chmielowska, Agata"https://www.zbmath.org/authors/?q=ai:chmielowska.agata"Różański, Michał"https://www.zbmath.org/authors/?q=ai:rozanski.michal"Smoleń, Barbara"https://www.zbmath.org/authors/?q=ai:smolen.barbara"Sobstyl, Ireneusz"https://www.zbmath.org/authors/?q=ai:sobstyl.ireneusz"Wituła, Roman"https://www.zbmath.org/authors/?q=ai:witula.romanSummary: The aim of this study was to present a simple method for finding the asymptotic relations for products of elements of some positive real sequences. The main reason to carry out this study was the result obtained by Alzer and Sandor concerning an estimation of a sequence of the product of the first \(k\) primes.Instances of computational optimal recovery: dealing with observation errorshttps://www.zbmath.org/1483.410122022-05-16T20:40:13.078697Z"Ettehad, Mahmood"https://www.zbmath.org/authors/?q=ai:ettehad.mahmood"Foucart, Simon"https://www.zbmath.org/authors/?q=ai:foucart.simonThe paper presents new contributions to optimizing functions recovery from observational data. The proposed approach relies on considering inaccurate data through some boundedness models and the emphasis is set on computational realization. The efficient construction of optimal recovery maps is discussed in several instances: local optimality under linearly or semidefinitely describable models, global optimality for the estimation of linear functionals under approximability models, and global near-optimality under approximability models in the space of continuous functions.
Reviewer: Sorin-Mihai Grad (Paris)Signal analysis using Born-Jordan-type distributionshttps://www.zbmath.org/1483.420042022-05-16T20:40:13.078697Z"Cordero, Elena"https://www.zbmath.org/authors/?q=ai:cordero.elena"de Gosson, Maurice"https://www.zbmath.org/authors/?q=ai:de-gosson.maurice-a"Dörfler, Monika"https://www.zbmath.org/authors/?q=ai:dorfler.monika"Nicola, Fabio"https://www.zbmath.org/authors/?q=ai:nicola.fabioThe Chapter contains results concerning recent advances in signal theory using time-frequency distributions. The authors demonstrate that some new members of the Cohen class that generalize the Wigner distribution are useful in damping artefacts in certain signals. Main properties as well as drawbacks are presented. Last but not least, several open problems are also discussed.
For the entire collection see [Zbl 1470.42002].
Reviewer: Liviu Goraş (Iaşi)Frames associated with shift invariant spaces on positive half linehttps://www.zbmath.org/1483.420172022-05-16T20:40:13.078697Z"Ahmad, Owais"https://www.zbmath.org/authors/?q=ai:ahmad.owais"Ahmad, Mobin"https://www.zbmath.org/authors/?q=ai:ahmad.mobin"Ahmad, Neyaz"https://www.zbmath.org/authors/?q=ai:ahmad.neyazThe authors show some necessary and sufficient conditions under which shift-invariant systems become frames on positive half line. Applications to Gabor frames and wavelet frames on positive half line are also presented.
Reviewer: Paşc Găvruţă (Timişoara)Equivalence of \(K\)-functionals and modulus of smoothness generated by a generalized Jacobi-Dunkl transform on the real linehttps://www.zbmath.org/1483.440032022-05-16T20:40:13.078697Z"Daher, Radouan"https://www.zbmath.org/authors/?q=ai:daher.radouan"Tyr, Othman"https://www.zbmath.org/authors/?q=ai:tyr.othmanSummary: Using the generalized Jacobi-Dunkl translation, we prove the equivalence between modulus of smoothness and \(K\)-functional constructed by the Sobolev space corresponding to the Jacobi-Dunkl Laplacian operator.Variational time discretization of Riemannian splineshttps://www.zbmath.org/1483.490172022-05-16T20:40:13.078697Z"Heeren, Behrend"https://www.zbmath.org/authors/?q=ai:heeren.behrend"Rumpf, Martin"https://www.zbmath.org/authors/?q=ai:rumpf.martin"Wirth, Benedikt"https://www.zbmath.org/authors/?q=ai:wirth.benediktSummary: We investigate a generalization of cubic splines to Riemannian manifolds. Spline curves are defined as minimizers of the spline energy -- a combination of the Riemannian path energy and the time integral of the squared covariant derivative of the path velocity -- under suitable interpolation conditions. A variational time discretization for the spline energy leads to a constrained optimization problem over discrete paths on the manifold. Existence of continuous and discrete spline curves is established using the direct method in the calculus of variations. Furthermore, the convergence of discrete spline paths to a continuous spline curve follows from the \(\Gamma \)-convergence of the discrete to the continuous spline energy. Finally, selected example settings are discussed, including splines on embedded finite-dimensional manifolds, on a high-dimensional manifold of discrete shells with applications in surface processing and on the infinite-dimensional shape manifold of viscous rods.Lagrangian manifolds and efficient short-wave asymptotics in a neighborhood of a caustic cusphttps://www.zbmath.org/1483.530942022-05-16T20:40:13.078697Z"Dobrokhotov, S. Yu."https://www.zbmath.org/authors/?q=ai:dobrokhotov.sergei-yu"Nazaikinskii, V. E."https://www.zbmath.org/authors/?q=ai:nazaikinskii.vladimir-eSummary: We develop an approach to writing efficient short-wave asymptotics based on the representation of the Maslov canonical operator in a neighborhood of generic caustics in the form of special functions of a composite argument. A constructive method is proposed that allows expressing the canonical operator near a caustic cusp corresponding to the Lagrangian singularity of type \(A_3\) (standard cusp) in terms of the Pearcey function and its first derivatives. It is shown that, conversely, the representation of a Pearcey type integral via the canonical operator turns out to be a very simple way to obtain its asymptotics for large real values of the arguments in terms of Airy functions and WKB-type functions.Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysishttps://www.zbmath.org/1483.650262022-05-16T20:40:13.078697Z"Sande, Espen"https://www.zbmath.org/authors/?q=ai:sande.espen"Manni, Carla"https://www.zbmath.org/authors/?q=ai:manni.carla"Speleers, Hendrik"https://www.zbmath.org/authors/?q=ai:speleers.hendrikIsogeometric analysis [\textit{T. J. R. Hughes} et al., Comput. Methods Appl. Mech. Eng. 194, No. 39--41, 4135--4195 (2005; Zbl 1151.74419)] is a generalization of the classical finite element analysis, and it advocates for the use of splines as finite elements. This has led to a renewed interest in using splines for approximation, and numerical evidence has shown the benefits of their smoothness -- smooth spline spaces demonstrate better approximation behaviour per degree of freedom than less smooth spline spaces.
The authors of the present manuscript provide a priori error estimates that, in particular, help to describe the afore-mentioned numerical evidence. The error estimates are for \(L^2\) and Ritz projections onto spline spaces of arbitrary smoothness defined on non-uniform meshes, and they feature constants that depend explicitly on the spline space parameters -- mesh size \(h\), polynomial degree \(p\) and smoothness \(k\). The key to their results is the description of the considered Sobolev spaces and spline spaces in terms of integral operators what allows them to derive explicit constants in spline approximation from those in polynomial approximation. The error estimates are provided for univariate spline spaces, reduced univariate spline spaces (i.e., spline spaces with zero end-point even or odd derivatives) and multivariate tensor-product spline spaces. The multivariate results cover approximation on the spline parametric domain as well as mapped single-patch and \(C^0\) multi-patch geometries; the role of the geometric map is explicitized in the constants. Finally, the error estimates improve upon the ones presented earlier in [\textit{L. Beirão da Veiga} et al., Numer. Math. 118, No. 2, 271--305 (2011; Zbl 1222.41010); \textit{S. Takacs} and \textit{T. Takacs}, Math. Models Methods Appl. Sci. 26, No. 7, 1411--1445 (2016; Zbl 1339.41012)] and [\textit{E. Sande} et al., Math. Models Methods Appl. Sci. 29, No. 6, 1175--1205 (2019; Zbl 1428.41010)].
Reviewer: Deepesh Toshniwal (Delft)The gap between theory and practice in function approximation with deep neural networkshttps://www.zbmath.org/1483.650282022-05-16T20:40:13.078697Z"Adcock, Ben"https://www.zbmath.org/authors/?q=ai:adcock.ben"Dexter, Nick"https://www.zbmath.org/authors/?q=ai:dexter.nick-cLasso hyperinterpolation over general regionshttps://www.zbmath.org/1483.650292022-05-16T20:40:13.078697Z"An, Congpei"https://www.zbmath.org/authors/?q=ai:an.congpei"Wu, Hao-Ning"https://www.zbmath.org/authors/?q=ai:wu.hao-ningTranslation matrix elements for spherical Gauss-Laguerre basis functionshttps://www.zbmath.org/1483.650352022-05-16T20:40:13.078697Z"Prestin, Jürgen"https://www.zbmath.org/authors/?q=ai:prestin.jurgen"Wülker, Christian"https://www.zbmath.org/authors/?q=ai:wulker.christianSummary: Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type \(L_{n-l-1}^{(l + 1/2)}(r^2) r^{l} Y_{lm}(\vartheta ,\varphi)\), \(|m| \le l < n \in{\mathbb{N}}\), constitute an orthonormal polynomial basis of the space \(L^{2}\) on \({\mathbb{R}}^{3}\) with radial Gaussian weight \(\exp (-r^{2})\). We have recently described reliable fast Fourier transforms for the SGL basis functions. The main application of the SGL basis functions and our fast algorithms is in solving certain three-dimensional rigid matching problems, where the center is prioritized over the periphery. For this purpose, so-called SGL translation matrix elements are required, which describe the spectral behavior of the SGL basis functions under translations. In this paper, we derive a closed-form expression of these translation matrix elements, allowing for a direct computation of these quantities in practice.Correction to: ``Translation matrix elements for spherical Gauss-Laguerre basis functions''https://www.zbmath.org/1483.650362022-05-16T20:40:13.078697Z"Prestin, Jürgen"https://www.zbmath.org/authors/?q=ai:prestin.jurgen"Wülker, Christian"https://www.zbmath.org/authors/?q=ai:wulker.christianCorrection to the authors' paper [ibid. 10, Paper No. 6, 16 p. (2019; Zbl 1483.65035), Formulae 3.6].Near-minimal cubature formulae on the diskhttps://www.zbmath.org/1483.650372022-05-16T20:40:13.078697Z"Benouahmane, Brahim"https://www.zbmath.org/authors/?q=ai:benouahmane.brahim"Annie, Cuyt"https://www.zbmath.org/authors/?q=ai:annie.cuyt"Yaman, Irem"https://www.zbmath.org/authors/?q=ai:yaman.iremSummary: The construction of (near-)minimal cubature formulae on the disk is still a complicated subject on which many results have been published. We restrict ourselves to the case of radial weight functions and make use of a recent connection between cubature and the concept of multivariate spherical orthogonal polynomials to derive a new system of equations defining the nodes and weights of (near-)minimal rules for general degree \(m=2n-1\), \(n \ge 2\). The approach encompasses all previous derivations.
The new system is small and may consist of only \((n+1)^2/4\) equations when \(n\) is odd and \(n(n+2)/4\) equations when \(n\) is even. It is valid for general \(n\) and has a Prony-like structure. It may admit a unique solution (such as for \(n=3)\) or an infinity of solutions (such as for \(n=7)\). In Section 2, the new approach is described, whereas the new system is derived in Sections 3 and 4. All well-known (near-)minimal cubature rules can be reobtained. Some typical illustrations of how this works are given in Section 5.
We expect that this unifying theory will shed new light on the topic of cubature, in particular with respect to the discovery of new bounds on the number of nodes and their connection with the zeros of multivariate orthogonal polynomials.Cubature formulae for the Gaussian weight. Some old and new rules.https://www.zbmath.org/1483.650422022-05-16T20:40:13.078697Z"Orive, Ramón"https://www.zbmath.org/authors/?q=ai:orive-rodriguez.ramon-angel"Santos-León, Juan C."https://www.zbmath.org/authors/?q=ai:santos-leon.juan-carlos"Spalević, Miodrag M."https://www.zbmath.org/authors/?q=ai:spalevic.miodrag-mThe authors consider the numerical evaluation of the integral
\[
\int_{\mathbb{R}^n}e^{-x^Tx}f(x)dx,
\]
where \(x=(x_1,\dots,x_n)\in \mathbb{R}^n\), and derive some new rules with respect to the aim of taking of certain degree of algebraic precision and a reasonable small number of nodes and good stability. The theory is presented by computing of new and classical methods of several numerical experiments.
Reviewer: Josef Kofroň (Praha)Gibbs phenomena for \(L^q\)-best approximation in finite element spaceshttps://www.zbmath.org/1483.651882022-05-16T20:40:13.078697Z"Houston, Paul"https://www.zbmath.org/authors/?q=ai:houston.paul"Roggendorf, Sarah"https://www.zbmath.org/authors/?q=ai:roggendorf.sarah"van der Zee, Kristoffer G."https://www.zbmath.org/authors/?q=ai:van-der-zee.kristoffer-georgeSummary: Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretization methods in non-standard function spaces, such as \(L^q\)-type Sobolev spaces. For \(q \rightarrow 1\), these methods have demonstrated huge potential in avoiding the notorious Gibbs phenomena, \textit{i.e.}, the occurrence of spurious non-physical oscillations near thin layers and jump discontinuities. In this work we provide theoretical results that explain some of these numerical observations. In particular, we investigate the Gibbs phenomena for \(L^q\)-best approximations of discontinuities in finite element spaces with \(1 \leq q < \infty\). We prove sufficient conditions on meshes in one and two dimensions such that over- and undershoots vanish in the limit \(q \rightarrow 1\). Moreover, we include examples of meshes such that Gibbs phenomena remain present even for \(q = 1\) and demonstrate that our results can be used to design meshes so as to eliminate the Gibbs phenomenon.Microscopic patterns in the 2D phase-field-crystal modelhttps://www.zbmath.org/1483.652052022-05-16T20:40:13.078697Z"Martine-La Boissonière, Gabriel"https://www.zbmath.org/authors/?q=ai:martine-la-boissoniere.gabriel"Choksi, Rustum"https://www.zbmath.org/authors/?q=ai:choksi.rustum"Lessard, Jean-Philippe"https://www.zbmath.org/authors/?q=ai:lessard.jean-philippeNew spectral element method for Volterra integral equations with weakly singular kernelhttps://www.zbmath.org/1483.652342022-05-16T20:40:13.078697Z"Zhang, Chao"https://www.zbmath.org/authors/?q=ai:zhang.chao.2|zhang.chao.7"Liu, Zhipeng"https://www.zbmath.org/authors/?q=ai:liu.zhipeng"Chen, Sheng"https://www.zbmath.org/authors/?q=ai:chen.sheng.2|chen.sheng.4|chen.sheng.3|chen.sheng.1"Tao, DongYa"https://www.zbmath.org/authors/?q=ai:tao.dongyaSummary: In this paper, we put forward a new spectral element method for the nonlinear second-kind Volterra integral equations (VIEs) with weakly singular kernel, which employs shifted Müntz-Jacobi functions and shifted Legendre polynomials as basis functions. This method is capable of approximating the limited regular solution more efficiently. We analyze the existence and uniqueness of the solution to the numerical scheme and derive the \(h p\)-version optimal convergence under some reasonable assumptions. A series of numerical examples are presented to demonstrate the efficiency of the new method.A new approach to solve the Schrodinger equation with an anharmonic sextic potentialhttps://www.zbmath.org/1483.810632022-05-16T20:40:13.078697Z"Nanni, Luca"https://www.zbmath.org/authors/?q=ai:nanni.lucaSummary: In this study, the N-dimensional radial Schrodinger equation with an anharmonic sextic potential is solved by the extended Nikirov-Uranov method. We prove that the radial function can be factorised as the product between an exponential function and a polynomial function solution of the biconfluent Heun equation. The approach investigated in this article aims to be an alternative to other known methods of solving, as it has the advantage of dealing with simple, first-order differential and algebraic equations and avoiding numerous and laborious coordinate transformations and series expansions.CCF approach for asymptotic option pricing under the CEV diffusionhttps://www.zbmath.org/1483.912562022-05-16T20:40:13.078697Z"Muroi, Yoshifumi"https://www.zbmath.org/authors/?q=ai:muroi.yoshifumiSummary: In the last two decades, the asymptotic expansion approach has become popular in mathematical finance because it enables us to obtain closed-form approximation formulae for many kinds of options within various kinds of financial models, such as local and stochastic volatility models. In this study, we propose an asymptotic expansion formula for the option price in a constant elasticity of variance model using the asymptotic expansion technique and Fourier analysis. This approach enables us to derive the higher order terms using only algebraic computation. Furthermore, this method enables us to derive not only the price of European options but also the price of options with an early exercise feature, such as Bermudan options and American options.