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Metric Approximation of Set-Valued Functions of Bounded Variation by Integral Operators Constr. Approx. (IF 2.7) Pub Date : 2024-03-13 Elena E. Berdysheva, Nira Dyn, Elza Farkhi, Alona Mokhov
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Non-polynomial q-Askey Scheme: Integral Representations, Eigenfunction Properties, and Polynomial Limits Constr. Approx. (IF 2.7) Pub Date : 2024-03-11
Abstract We construct a non-polynomial generalization of the q-Askey scheme. Whereas the elements of the q-Askey scheme are given by q-hypergeometric series, the elements of the non-polynomial scheme are given by contour integrals, whose integrands are built from Ruijsenaars’ hyperbolic gamma function. Alternatively, the integrands can be expressed in terms of Faddeev’s quantum dilogarithm, Woronowicz’s
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Asymptotics of Chebyshev Rational Functions with Respect to Subsets of the Real Line Constr. Approx. (IF 2.7) Pub Date : 2024-03-09 Benjamin Eichinger, Milivoje Lukić, Giorgio Young
There is a vast theory of Chebyshev and residual polynomials and their asymptotic behavior. The former ones maximize the leading coefficient and the latter ones maximize the point evaluation with respect to an \(L^\infty \) norm. We study Chebyshev and residual extremal problems for rational functions with real poles with respect to subsets of \(\overline{{{\mathbb {R}}}}\). We prove root asymptotics
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Optimal Rates of Approximation by Shallow ReLU $$^k$$ Neural Networks and Applications to Nonparametric Regression Constr. Approx. (IF 2.7) Pub Date : 2024-02-26
Abstract We study the approximation capacity of some variation spaces corresponding to shallow ReLU \(^k\) neural networks. It is shown that sufficiently smooth functions are contained in these spaces with finite variation norms. For functions with less smoothness, the approximation rates in terms of the variation norm are established. Using these results, we are able to prove the optimal approximation
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Sharp Bernstein Inequalities on Simplex Constr. Approx. (IF 2.7) Pub Date : 2024-02-24 Yan Ge, Yuan Xu
We prove several new families of Bernstein inequalities of two types on the simplex. The first type consists of inequalities in \(L^2\) norm for the Jacobi weight, some of which are sharp, and they are established via the spectral operator that has orthogonal polynomials as eigenfunctions. The second type consists of inequalities in \(L^p\) norm for doubling weight on the simplex. The first type is
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Dirac Operators with Exponentially Decaying Entropy Constr. Approx. (IF 2.7) Pub Date : 2024-02-21 Pavel Gubkin
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An Extremal Problem for the Bergman Kernel of Orthogonal Polynomials Constr. Approx. (IF 2.7) Pub Date : 2024-01-30 S. Charpentier, N. Levenberg, F. Wielonsky
Let \(\Gamma \subset \mathbb {C}\) be a curve of class \(C(1,\alpha )\). For \(z_{0}\) in the unbounded component of \(\mathbb {C}\setminus \Gamma \), and for \(n=1,2,...\), let \(\nu _n\) be a probability measure with \(\mathop {\textrm{supp}}\nolimits (\nu _{n})\subset \Gamma \) which minimizes the Bergman function \(B_{n}(\nu ,z):=\sum _{k=0}^{n}|q_{k}^{\nu }(z)|^{2}\) at \(z_{0}\) among all probability
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Loewner Theory for Bernstein Functions I: Evolution Families and Differential Equations Constr. Approx. (IF 2.7) Pub Date : 2024-01-22 Pavel Gumenyuk, Takahiro Hasebe, José-Luis Pérez
One-parameter semigroups of holomorphic functions appear naturally in various applications of Complex Analysis, and in particular, in the theory of (temporally) homogeneous branching processes. A suitable analogue of one-parameter semigroups in the inhomogeneous setting is the notion of a (reverse) evolution family. In this paper we study evolution families formed by Bernstein functions, which play
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On the Definition of Higher Gamma Functions Constr. Approx. (IF 2.7) Pub Date : 2024-01-20 Ricardo Pérez-Marco
We extent our definition of Euler Gamma function to higher Gamma functions, and we give a unified characterization of Barnes higher Gamma functions, Mellin Gamma functions, Barnes multiple Gamma functions, Jackson q-Gamma function, and Nishizawa higher q-Gamma functions in the space of finite order meromorphic functions. The method extends to more general functional equations and unveils the multiplicative
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Maximal Function and Riesz Transform Characterizations of Hardy Spaces Associated with Homogeneous Higher Order Elliptic Operators and Ball Quasi-Banach Function Spaces Constr. Approx. (IF 2.7) Pub Date : 2024-01-09 Xiaosheng Lin, Dachun Yang, Sibei Yang, Wen Yuan
Let L be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on \({\mathbb {R}}^n\) and X a ball quasi-Banach function space on \({\mathbb {R}}^n\) satisfying some mild assumptions. Denote by \(H_{X,\, L}({\mathbb {R}}^n)\) the Hardy space, associated with both L and X, which is defined via the Lusin area function related to the semigroup generated
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Complete Minimal Logarithmic Energy Asymptotics for Points in a Compact Interval: A Consequence of the Discriminant of Jacobi Polynomials Constr. Approx. (IF 2.7) Pub Date : 2023-12-30 J. S. Brauchart
The electrostatic interpretation of zeros of Jacobi polynomials, due to Stieltjes and Schur, enables us to obtain the complete asymptotic expansion as \(n \rightarrow \infty \) of the minimal logarithmic potential energy of n point charges restricted to move in the interval \([-1,1]\) in the presence of an external field generated by endpoint charges. By the same methods, we determine the complete
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Bounds on Orthonormal Polynomials for Restricted Measures Constr. Approx. (IF 2.7) Pub Date : 2023-12-18 D. S. Lubinsky
Suppose that \(\nu \) is a given positive measure on \(\left[ -1,1\right] \), and that \(\mu \) is another measure on the real line, whose restriction to \( \left( -1,1\right) \) is \(\nu \). We show that one can bound the orthonormal polynomials \(p_{n}\left( \mu ,y\right) \) for \(\mu \) and \(y\in \mathbb {R}\), by the supremum of \(\left| S_{J}\left( y\right) p_{n-J}\left( S_{J}^{2}\nu ,y\right)
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Monotone Discretization of Anisotropic Differential Operators Using Voronoi’s First Reduction Constr. Approx. (IF 2.7) Pub Date : 2023-12-01 Frédéric Bonnans, Guillaume Bonnet, Jean-Marie Mirebeau
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Applications of the Lipschitz Summation Formula and a Generalization of Raabe’s Cosine Transform Constr. Approx. (IF 2.7) Pub Date : 2023-10-24 Atul Dixit, Rahul Kumar
General summation formulas have been proved to be very useful in analysis, number theory and other branches of mathematics. The Lipschitz summation formula is one of them. In this paper, we give its application by providing a new transformation formula which generalizes that of Ramanujan. Ramanujan’s result, in turn, is a generalization of the modular transformation of Eisenstein series \(E_k(z)\)
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Polynomial Approximation on $$C^2$$ -Domains Constr. Approx. (IF 2.7) Pub Date : 2023-10-21 Feng Dai, Andriy Prymak
We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact \(C^2\)-domain \(\Omega \subset \mathbb {R}^d\). This new modulus of smoothness is defined via finite differences along the directions of coordinate axes, and along a number of tangential directions from the boundary. With this modulus, we prove
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Shrinking Schauder Frames and Their Associated Bases Constr. Approx. (IF 2.7) Pub Date : 2023-10-17 Kevin Beanland, Daniel Freeman
For a Banach space X with a shrinking Schauder frame \((x_i,f_i)\) we provide an explicit method for constructing a shrinking associated basis. In the case that the minimal associated basis is not shrinking, we prove that every shrinking associated basis of \((x_i,f_i)\) dominates an uncountable family of incomparable shrinking associated bases of \((x_i,f_i)\). By adapting a construction of Pełczyński
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Stable Gabor Phase Retrieval in Gaussian Shift-Invariant Spaces via Biorthogonality Constr. Approx. (IF 2.7) Pub Date : 2023-10-04 Philipp Grohs, Lukas Liehr
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On the Log-Concavity of the Wright Function Constr. Approx. (IF 2.7) Pub Date : 2023-08-24 Rui A. C. Ferreira, Thomas Simon
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Existence of Almost Greedy Bases in Mixed-Norm Sequence and Matrix Spaces, Including Besov Spaces Constr. Approx. (IF 2.7) Pub Date : 2023-07-29 Fernando Albiac, José L. Ansorena, Glenier Bello, Przemysław Wojtaszczyk
We prove that the sequence spaces \(\ell _p\oplus \ell _q\) and the spaces of infinite matrices \(\ell _p(\ell _q)\), \(\ell _q(\ell _p)\) and \((\bigoplus _{n=1}^\infty \ell _p^n)_{\ell _q}\), which are isomorphic to certain Besov spaces, have an almost greedy basis whenever \(0
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The WCGA in $$L^p(\log L)^{\alpha }$$ Spaces Constr. Approx. (IF 2.7) Pub Date : 2023-07-25 Gustavo Garrigós
We present some new results concerning Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces \({{\mathbb {X}}}\). First, we generalize Temlyakov’s theorem (Temlyakov in Forum Math Sigma 2(12):26, 2014) to cover situations in which the modulus of smoothness and the \({\texttt {A3}}\) parameter are not necessarily power functions. Secondly, we apply
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Extension Operators for Some Ultraholomorphic Classes Defined by Sequences of Rapid Growth Constr. Approx. (IF 2.7) Pub Date : 2023-07-15 Javier Jiménez-Garrido, Alberto Lastra, Javier Sanz
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Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere Constr. Approx. (IF 2.7) Pub Date : 2023-07-13 Sergiy Borodachov
We consider the problem of finding an N-point configuration on the sphere \(S^d\subset \mathbb R^{d+1}\) with the smallest absolute maximum value over \(S^d\) of its total potential. The potential induced by each point \(\textbf{y}\) in a given configuration at a point \(\textbf{x}\in S^d\) is \(f\left( \left| \textbf{x}-\textbf{y}\right| ^2\right) \), where f is continuous on [0, 4] and completely
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Modulated Bi-Orthogonal Polynomials on the Unit Circle: The $$2j-k$$ and $$j-2k$$ Systems Constr. Approx. (IF 2.7) Pub Date : 2023-06-21 Roozbeh Gharakhloo, Nicholas S. Witte
We construct the systems of bi-orthogonal polynomials on the unit circle where the Toeplitz structure of the moment determinants is replaced by \(\det (w_{2j-k})_{0\le j,k \le N-1} \) and the corresponding Vandermonde modulus squared is replaced by \(\prod _{1 \le j < k \le N}(\zeta _k - \zeta _j)(\zeta ^{-2}_k - \zeta ^{-2}_j) \). This is the simplest case of a general system of \(pj-qk\) with p, q
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On the Omnipresence of Spurious Local Minima in Certain Neural Network Training Problems Constr. Approx. (IF 2.7) Pub Date : 2023-06-14 Constantin Christof, Julia Kowalczyk
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A lower bound for the dimension of tetrahedral splines in large degree Constr. Approx. (IF 2.7) Pub Date : 2023-06-12 Michael DiPasquale, Nelly Villamizar
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Rodrigues’ Descendants of a Polynomial and Boutroux Curves Constr. Approx. (IF 2.7) Pub Date : 2023-05-30 Rikard Bøgvad, Christian Hägg, Boris Shapiro
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Recursion Formulas for Integrated Products of Jacobi Polynomials Constr. Approx. (IF 2.7) Pub Date : 2023-05-27 Sven Beuchler, Tim Haubold, Veronika Pillwein
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On the Fourier Coefficients of Powers of a Blaschke Factor and Strongly Annular Functions Constr. Approx. (IF 2.7) Pub Date : 2023-05-25 Alexander Borichev, Karine Fouchet, Rachid Zarouf
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Spectral Properties of Sierpinski Measures on $$\mathbb {R}^n$$ Constr. Approx. (IF 2.7) Pub Date : 2023-05-21 Xin-Rong Dai, Xiao-Ye Fu, Zhi-Hui Yan
Let \(R=\varrho I_{n}\) and \(\mathcal {D}=\left\{ \textbf{0},\textbf{e}_{1},\ldots ,\textbf{e}_{n}\right\} \), where \(\varrho >1\) and \(\textbf{e}_{i}\) is the i-th coordinate vector in \(\mathbb {R}^n\). The spectral properties of the \(n-\)dimensional Sierpinski measure \(\mu _{R,\mathcal {D}}\) has been studied over two decades. In this paper, a special type of spectrum called a typical spectrum
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Orthogonal Polynomials with Periodically Modulated Recurrence Coefficients in the Jordan Block Case II Constr. Approx. (IF 2.7) Pub Date : 2023-05-18 Grzegorz Świderski, Bartosz Trojan
We study Jacobi matrices with N-periodically modulated recurrence coefficients when the sequence of N-step transfer matrices is convergent to a non-trivial Jordan block. In particular, we describe asymptotic behavior of their generalized eigenvectors, we prove convergence of N-shifted Turán determinants as well as of the Christoffel–Darboux kernel on the diagonal. Finally, by means of subordinacy theory
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On Extremal Functions and V. Markov Type Polynomial Inequality for Certain Subsets of $${\mathbb {R}}^N$$ Constr. Approx. (IF 2.7) Pub Date : 2023-05-15 Mirosław Baran, Grzegorz Sroka
We introduce a polynomial extremal function \(\Phi (E,{\mathbb {F}},z)\) which is one of possible generalizations of the classical Siciak extremal function, restricted to subspaces \({\mathbb {F}}\) of the linear space of all polynomials of N variables that are invariant under differentiation. We show that the so-called HCP condition in this situation: \(\log \Phi (E,{\mathbb {F}},z)\le A\text { dist}(z
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Estimates for the Largest Critical Value of $$T_n^{(k)}$$ Constr. Approx. (IF 2.7) Pub Date : 2023-05-14 Nikola Naidenov, Geno Nikolov
For \(T_n(x)=\cos n\arccos x\), \(x\in [-1,1]\), the n-th Chebyshev polynomial of the first kind, we study the quantity $$\begin{aligned} \tau _{n,k}:=\frac{|T_n^{(k)}(\omega _{n,k})|}{T_n^{(k)}(1)},\quad 1\le k\le n-2, \end{aligned}$$ where \(T_n^{(k)}\) is the k-th derivative of \(T_n\) and \(\omega _{n,k}\) is the largest zero of \(T_n^{(k+1)}\). Since the absolute values of the local extrema of
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Set-Valued $$\alpha $$ -Fractal Functions Constr. Approx. (IF 2.7) Pub Date : 2023-05-13 Megha Pandey, Tanmoy Som, Saurabh Verma
In this paper, we introduce the concept of the \(\alpha \)-fractal function and fractal approximation for a set-valued continuous map defined on a closed and bounded interval of real numbers. Also, we study some properties of such fractal functions. Further, we estimate the perturbation error between the given continuous function and its \(\alpha \)-fractal function. Additionally, we define a new graph
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A Malmquist–Steinmetz Theorem for Difference Equations Constr. Approx. (IF 2.7) Pub Date : 2023-05-03 Yueyang Zhang, Risto Korhonen
It is shown that if the equation $$\begin{aligned} f(z+1)^n=R(z,f), \end{aligned}$$ where R(z, f) is rational in both arguments and \(\deg _f(R(z,f))\not =n\), has a transcendental meromorphic solution, then the equation above reduces into one out of several types of difference equations where the rational term R(z, f) takes particular forms. Solutions of these equations are presented in terms of Weierstrass
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The spherical ensemble and quasi-Monte-Carlo designs Constr. Approx. (IF 2.7) Pub Date : 2023-04-29 Robert J. Berman
The spherical ensemble is a well-known ensemble of N repulsive points on the two-dimensional sphere, which can realized in various ways (as a random matrix ensemble, a determinantal point process, a Coulomb gas, a Quantum Hall state...). Here we show that the spherical ensemble enjoys remarkable convergence properties from the point of view of numerical integration. More precisely, it is shown that
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Uniform Asymptotic Expansions for Gegenbauer Polynomials and Related Functions via Differential Equations Having a Simple Pole Constr. Approx. (IF 2.7) Pub Date : 2023-04-29 T. M. Dunster
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Minimum Energy Problems with External Fields on Locally Compact Spaces Constr. Approx. (IF 2.7) Pub Date : 2023-04-25 Natalia Zorii
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Universal Sampling Discretization Constr. Approx. (IF 2.7) Pub Date : 2023-04-25 F. Dai, V. Temlyakov
Let \(X_N\) be an N-dimensional subspace of \(L_2\) functions on a probability space \((\Omega , \mu )\) spanned by a uniformly bounded Riesz basis \(\Phi _N\). Given an integer \(1\le v\le N\) and an exponent \(1\le p\le 2\), we obtain universal discretization for the integral norms \(L_p(\Omega ,\mu )\) of functions from the collection of all subspaces of \(X_N\) spanned by v elements of \(\Phi _N\)
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Radial Basis Function Approximation with Distributively Stored Data on Spheres Constr. Approx. (IF 2.7) Pub Date : 2023-04-24 Han Feng, Shao-Bo Lin, Ding-Xuan Zhou
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Path Regularity of the Brownian Motion and the Brownian Sheet Constr. Approx. (IF 2.7) Pub Date : 2023-04-19 H. Kempka, C. Schneider, J. Vybiral
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A Lower Bound for the Logarithmic Energy on $$\mathbb {S}^2$$ and for the Green Energy on $$\mathbb {S}^n$$ Constr. Approx. (IF 2.7) Pub Date : 2023-04-19 Carlos Beltrán, Fátima Lizarte
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Asymptotics of k-nearest Neighbor Riesz Energies Constr. Approx. (IF 2.7) Pub Date : 2023-04-15 Douglas P. Hardin, Edward B. Saff, Oleksandr Vlasiuk
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Richard S. Varga (October 9, 1928 -February 25, 2022) Constr. Approx. (IF 2.7) Pub Date : 2023-04-10 Edward B. Saff
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Special issue of Constructive Approximation dedicated to Ronald A. DeVore Constr. Approx. (IF 2.7) Pub Date : 2023-04-11 Edward B. Saff
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Non-Hermitian Orthogonal Polynomials on a Trefoil Constr. Approx. (IF 2.7) Pub Date : 2023-04-06 Ahmad B. Barhoumi, Maxim L. Yattselev
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Spectral Decomposition of Discrepancy Kernels on the Euclidean Ball, the Special Orthogonal Group, and the Grassmannian Manifold Constr. Approx. (IF 2.7) Pub Date : 2023-04-07 Josef Dick, Martin Ehler, Manuel Gräf, Christian Krattenthaler
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Duality and Difference Operators for Matrix Valued Discrete Polynomials on the Nonnegative Integers Constr. Approx. (IF 2.7) Pub Date : 2023-03-28 Bruno Eijsvoogel, Lucía Morey, Pablo Román
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Universal Regular Conditional Distributions via Probabilistic Transformers Constr. Approx. (IF 2.7) Pub Date : 2023-03-27 Anastasis Kratsios
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Dominating Sets in Bergman Spaces on Strongly Pseudoconvex Domains Constr. Approx. (IF 2.7) Pub Date : 2023-03-24 A. Walton Green, Nathan A. Wagner
We obtain local estimates, also called propagation of smallness or Remez-type inequalities, for analytic functions in several variables. Using Carleman estimates, we obtain a three sphere-type inequality, where the outer two spheres can be any sets satisfying a boundary separation property, and the inner sphere can be any set of positive Lebesgue measure. We apply this local result to characterize
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New Orthogonality Relations for Super-Jack Polynomials and an Associated Lassalle–Nekrasov Correspondence Constr. Approx. (IF 2.7) Pub Date : 2023-03-17 Martin Hallnäs
The super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski, are polynomials in \(n+m\) variables, which reduce to the Jack polynomials when \(n=0\) or \(m=0\) and provide joint eigenfunctions of the quantum integrals of the deformed trigonometric Calogero–Moser–Sutherland system. We prove that the super-Jack polynomials are orthogonal with respect to a bilinear form of the form \((p,q)\mapsto
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Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics Constr. Approx. (IF 2.7) Pub Date : 2023-03-15 Mario Kieburg, Shi-Hao Li, Jiyuan Zhang, Peter J. Forrester
A framework to study the eigenvalue probability density function for products of unitary random matrices with an invariance property is developed. This involves isolating a class of invariant unitary matrices, to be referred to as cyclic Pólya ensembles, and examining their properties with respect to the spherical transform on \(\mathrm U(N)\). Included in the cyclic Pólya ensemble class are Haar invariant
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Wiener–Hopf Difference Equations and Semi-Cardinal Interpolation with Integrable Convolution Kernels Constr. Approx. (IF 2.7) Pub Date : 2023-03-11 Aurelian Bejancu
Let \(H\subset {\mathbb {Z}}^d\) be a half-space lattice, defined either relative to a fixed coordinate (e.g. \(H = {\mathbb {Z}}^{d-1}\!\times \!{\mathbb {Z}}_+\)), or relative to a linear order \(\preceq \) on \({\mathbb {Z}}^d\), i.e. \(H = \{j\in {\mathbb {Z}}^d: 0\preceq j\}\). We consider the problem of interpolation at the points of H from the space of series expansions in terms of the H-shifts
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Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions Constr. Approx. (IF 2.7) Pub Date : 2023-03-08 Mazen Ali, Anthony Nouy
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The Product of m Real $$N\times N$$ Ginibre Matrices: Real Eigenvalues in the Critical Regime $$m=O(N)$$ Constr. Approx. (IF 2.7) Pub Date : 2023-03-02 Gernot Akemann, Sung-Soo Byun
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Sobolev-Type Embeddings for Neural Network Approximation Spaces Constr. Approx. (IF 2.7) Pub Date : 2023-03-05 Philipp Grohs, Felix Voigtlaender
We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated (with error measured in \(L^p\)) by ReLU neural networks with an increasing number of coefficients, subject to bounds on the magnitude of the coefficients and the number of hidden layers. We prove embedding theorems between these spaces for different values of p. Furthermore
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Probabilistic and Analytical Aspects of the Symmetric and Generalized Kaiser–Bessel Window Function Constr. Approx. (IF 2.7) Pub Date : 2023-03-02 Árpád Baricz, Tibor K. Pogány
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Characterization of the Variation Spaces Corresponding to Shallow Neural Networks Constr. Approx. (IF 2.7) Pub Date : 2023-02-22 Jonathan W. Siegel, Jinchao Xu
We study the variation space corresponding to a dictionary of functions in \(L^2(\Omega )\) for a bounded domain \(\Omega \subset {\mathbb {R}}^d\). Specifically, we compare the variation space, which is defined in terms of a convex hull with related notions based on integral representations. This allows us to show that three important notions relating to the approximation theory of shallow neural
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Multivariate Generalized Hermite Subdivision Schemes Constr. Approx. (IF 2.7) Pub Date : 2023-02-14 Bin Han
Due to properties such as interpolation, smoothness, and spline connections, Hermite subdivision schemes employ fast iterative algorithms for geometrically modeling curves/surfaces in CAGD and for building Hermite wavelets in numerical PDEs. In this paper, we introduce a notion of generalized Hermite (dyadic) subdivision schemes and then we characterize their convergence, smoothness and underlying
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Sparse Estimation: An MMSE Approach Constr. Approx. (IF 2.7) Pub Date : 2023-02-14 Tongyao Pang, Zuowei Shen
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The Spectrality of Self-affine Measure Under the Similar Transformation of $$GL_n(p)$$ Constr. Approx. (IF 2.7) Pub Date : 2023-02-03 Jing-Cheng Liu, Zhi-Yong Wang
Let \(\mu _{M,D}\) be the self-affine measure generated by an expanding integer matrix \(M\in M_n({\mathbb {Z}})\) and a finite digit set \(D\subset {\mathbb {Z}}^n\). It is well known that the two measures \(\mu _{M,D}\) and \(\mu _{\widetilde{M},\widetilde{D}}\) have the same spectrality if \({\widetilde{M}}=B^{-1}MB\) and \({\widetilde{D}}=B^{-1}D\), where \(B\in M_n({\mathbb {R}})\) is a nonsingular