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On Baker’s Patchwork Conjecture for Diagonal Padé Approximants Constr. Approx. (IF 1.531) Pub Date : 2021-01-05 D. S. Lubinsky
We prove that for entire functions f of finite order, there is a sequence of integers \(\mathcal {S}\) such that as \(n\rightarrow \infty \) through S, $$\begin{aligned} \min \left\{ \left| f-\left[ n/n\right] \right| \left( z\right) ,\left| f-\left[ n-1/n-1\right] \right| \left( z\right) \right\} ^{1/n}\rightarrow 0 \end{aligned}$$ uniformly for z in compact subsets of the plane. More generally this
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Orthogonal Polynomials with Ultra-Exponential Weight Functions: An Explicit Solution to the Ditkin–Prudnikov Problem Constr. Approx. (IF 1.531) Pub Date : 2021-01-03 S. Yakubovich
New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, we give an explicit solution to the Ditkin–Prudnikov problem (1966). The 3-term recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are derived. The method is based on differential properties of the involved special functions and their representations
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Some q -Supercongruences from Transformation Formulas for Basic Hypergeometric Series Constr. Approx. (IF 1.531) Pub Date : 2020-11-30 Victor J. W. Guo, Michael J. Schlosser
Several new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Zudilin. More concretely, the results in this paper include q-analogues of supercongruences (referring to p-adic identities remaining
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Optimal Polynomial Prediction Measures and Extremal Polynomial Growth Constr. Approx. (IF 1.531) Pub Date : 2020-11-02 L. Bos, N. Levenberg, J. Ortega-Cerdà
We show that the problem of finding the measure supported on a compact set \(K\subset \mathbb {C}\) such that the variance of the least squares predictor by polynomials of degree at most n at a point \(z_0\in \mathbb {C}^d\backslash K\) is a minimum is equivalent to the problem of finding the polynomial of degree at most n, bounded by 1 on K, with extremal growth at \(z_0.\) We use this to find the
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Variable Besov Spaces Associated with Heat Kernels Constr. Approx. (IF 1.531) Pub Date : 2020-10-15 Ciqiang Zhuo, Dachun Yang
Let \(({\mathcal {X}},\rho , \mu )\) be a space of homogeneous type. Suppose that \(p(\cdot ),\ q(\cdot ):\ {\mathcal {X}}\rightarrow (0,\infty ]\) are such that both \(1/p(\cdot )\) and \(1/q(\cdot )\) satisfy the globally log-Hölder continuous condition, and \(s(\cdot ):\ {\mathcal {X}}\rightarrow \mathbb R\) is a bounded function satisfying the locally log-Hölder continuous condition. In this article
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Asymptotic Behaviour of Christoffel–Darboux Kernel Via Three-Term Recurrence Relation I Constr. Approx. (IF 1.531) Pub Date : 2020-09-23 Grzegorz Świderski, Bartosz Trojan
For Jacobi parameters belonging to one of three classes: asymptotically periodic, periodically modulated, and the blend of these two, we study the asymptotic behavior of the Christoffel functions and the scaling limits of the Christoffel–Darboux kernel. We assume regularity of Jacobi parameters in terms of the Stolz class. We emphasize that the first class only gives rise to measures with compact supports
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Logarithmic Equilibrium on the Sphere in the Presence of Multiple Point Charges Constr. Approx. (IF 1.531) Pub Date : 2020-09-21 A. R. Legg, P. D. Dragnev
With the sphere \(\mathbb {S}^2 \subset \mathbb {R}^3\) as a conductor holding a unit charge with logarithmic interactions, we consider the problem of determining the support of the equilibrium measure in the presence of an external field consisting of finitely many point charges on the surface of the sphere. We determine that for any such configuration, the complement of the equilibrium support is
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Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane Constr. Approx. (IF 1.531) Pub Date : 2020-09-18 Gernot Akemann, Taro Nagao, Iván Parra, Graziano Vernizzi
We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer polynomials \(C_n^{(1+\alpha )}(z)\) for \(\alpha >-1\) containing the Legendre polynomials \(P_n(z)\) and the subset \(P_n^{(\alpha +\frac{1}{2},\pm \frac{1}{2})}(z)\)
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Distortion and Critical Values of the Finite Blaschke Product Constr. Approx. (IF 1.531) Pub Date : 2020-09-18 V. N. Dubinin
We establish a sharp upper bound for the absolute value of the derivative of the finite Blaschke product, provided that the critical values of this product lie in a given disk.
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Kesten–McKay Law for Random Subensembles of Paley Equiangular Tight Frames Constr. Approx. (IF 1.531) Pub Date : 2020-09-15 Mark Magsino, Dustin G. Mixon, Hans Parshall
We apply the method of moments to prove a recent conjecture of Haikin, Zamir and Gavish concerning the distribution of the singular values of random subensembles of Paley equiangular tight frames. Our analysis applies more generally to real equiangular tight frames of redundancy 2, and we suspect similar ideas will eventually produce more general results for arbitrary choices of redundancy.
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Cubature Rules for Unitary Jacobi Ensembles Constr. Approx. (IF 1.531) Pub Date : 2020-08-04 J. F. van Diejen, E. Emsiz
We present Chebyshev type cubature rules for the exact integration of rational symmetric functions with poles on prescribed coordinate hyperplanes. Here the integration is with respect to the densities of unitary Jacobi ensembles stemming from the Haar measures of the orthogonal and the compact symplectic Lie groups.
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Entropy Numbers of Finite Dimensional Mixed-Norm Balls and Function Space Embeddings with Small Mixed Smoothness Constr. Approx. (IF 1.531) Pub Date : 2020-08-03 Sebastian Mayer, Tino Ullrich
We study the embedding \(\mathrm {id}: \ell _p^b(\ell _q^d) \rightarrow \ell _r^b(\ell _u^d)\) and prove matching bounds for the entropy numbers \(e_k(\mathrm {id})\) provided that \(0
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Hyperuniform Point Sets on Flat Tori: Deterministic and Probabilistic Aspects Constr. Approx. (IF 1.531) Pub Date : 2020-07-21 Tetiana A. Stepanyuk
In this paper we study hyperuniformity on flat tori. Hyperuniform point sets on the unit sphere have been studied by J. Brauchart, P. Grabner, W. Kusner and J. Ziefle. It is shown that point sets which are hyperuniform for large balls, small balls, or balls of threshold order on the flat tori are uniformly distributed. Moreover, it is also shown that QMC-designs sequences for Sobolev classes, probabilistic
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Polynomial Approximation of Anisotropic Analytic Functions of Several Variables Constr. Approx. (IF 1.531) Pub Date : 2020-07-16 Andrea Bonito, Ronald DeVore, Diane Guignard, Peter Jantsch, Guergana Petrova
Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on Taylor expansions, and study their approximation by finite dimensional polynomial spaces \({{\mathcal {P}}}_\Lambda \) described by lower sets \(\Lambda \). Given
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Zero Distribution of Orthogonal Polynomials on a q -Lattice Constr. Approx. (IF 1.531) Pub Date : 2020-07-06 Walter Van Assche, Quinten Van Baelen
We give the asymptotic behavior of the zeros of orthogonal polynomials, after appropriate scaling, for which the orthogonality measure is supported on the q-lattice \(\{q^k, k=0,1,2,3,\ldots \}\), where \(0< q < 1\). The asymptotic distribution of the zeros is given by the radial part of the equilibrium measure of an extremal problem in logarithmic potential theory for circular symmetric measures with
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Turán-Type Reverse Markov Inequalities for Polynomials with Restricted Zeros Constr. Approx. (IF 1.531) Pub Date : 2020-06-05 Tamás Erdélyi
Let \(\mathcal{P}_n^c\) denote the set of all algebraic polynomials of degree at most n with complex coefficients. Let $$\begin{aligned} D^+ := \{z \in \mathbb {C}: |z| \le 1, \text { Im}(z) \ge 0\}\,. \end{aligned}$$ For integers \(0 \le k \le n\) let \(\mathcal{F}_{n,k}^c\) be the set of all polynomials \(P \in \mathcal{P}_n^c\) having at least \(n-k\) zeros in \(D^+\). Let $$\begin{aligned} \Vert
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Boundary Behavior of Optimal Polynomial Approximants Constr. Approx. (IF 1.531) Pub Date : 2020-05-21 Catherine Bénéteau, Myrto Manolaki, Daniel Seco
In this paper, we provide an efficient method for computing the Taylor coefficients of \(1-p_n f\), where \(p_n\) denotes the optimal polynomial approximant of degree n to 1/f in a Hilbert space \(H^2_\omega \) of analytic functions over the unit disc \(\mathbb {D}\), and f is a polynomial of degree d with d simple zeros. As a consequence, we show that in many of the spaces \(H^2_\omega \), the sequence
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Rational Minimax Iterations for Computing the Matrix p th Root Constr. Approx. (IF 1.531) Pub Date : 2020-05-12 Evan S. Gawlik
In a previous paper by the author, a family of iterations for computing the matrix square root was constructed by exploiting a recursion obeyed by Zolotarev’s rational minimax approximants of the function \(z^{1/2}\). The present paper generalizes this construction by deriving rational minimax iterations for the matrix pth root, where \(p \ge 2\) is an integer. The analysis of these iterations is considerably
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Hermite-Poulain Theorems for Linear Finite Difference Operators Constr. Approx. (IF 1.531) Pub Date : 2020-05-12 Olga Katkova, Mikhail Tyaglov, Anna Vishnyakova
We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form $$\begin{aligned} \Delta _{\theta , h}(f)(z)=e^{i\theta }f(z+ih)-e^{-i\theta }f(z-ih), \quad \theta \in [0,\pi ),\ \ h\in
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Approximation to uniform distribution in $$\mathrm {SO}(3)$$SO(3) Constr. Approx. (IF 1.531) Pub Date : 2020-05-09 Carlos Beltrán, Damir Ferizović
Using the theory of determinantal point processes we give upper bounds for the Green and Riesz energies for the rotation group \(\mathrm {SO}(3)\), with Riesz parameter up to 3. The Green function is computed explicitly, and a lower bound for the Green energy is established, enabling comparison of uniform point constructions on \(\mathrm {SO}(3)\). The variance of rotation matrices sampled by a certain
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On a Uniqueness Theorem of E. B. Vul Constr. Approx. (IF 1.531) Pub Date : 2020-04-27 Sasha Sodin
We recall a uniqueness theorem of E. B. Vul pertaining to a version of the cosine transform originating in spectral theory. Then we point out an application to the Bernstein approximation problem with non-symmetric weights: a theorem of Volberg is proved by elementary means.
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Planar Sampling Sets for the Short-Time Fourier Transform Constr. Approx. (IF 1.531) Pub Date : 2020-04-09 Philippe Jaming, Michael Speckbacher
This paper considers the problem of restricting the short-time Fourier transform to sets of nonzero measure in the plane. Thereby, we study under which conditions one has a sampling set and provide estimates of the corresponding sampling bound. In particular, we give a quantitative estimate for the lower sampling bound in the case of Hermite windows and derive a sufficient condition for a large class
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Ambient Approximation on Embedded Submanifolds Constr. Approx. (IF 1.531) Pub Date : 2020-03-09 L.-B. Maier
In this paper, we present a generalization of Lehmann’s approach for solving approximation problems on hypersurfaces to situations with arbitrary codimension. We show that as in the case of hypersurfaces, the method is able to transfer approximation orders from the ambient space to the submanifold. In particular, the resulting approximant is \({\mathrm {C}}^{m-2}\) and the error decays at an optimal
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The Generalized Operator Based Prony Method Constr. Approx. (IF 1.531) Pub Date : 2020-02-27 Kilian Stampfer, Gerlind Plonka
The generalized Prony method is a reconstruction technique for a large variety of sparse signal models that can be represented as sparse expansions into eigenfunctions of a linear operator A. However, this procedure requires the evaluation of higher powers of the linear operator A that are often expensive to provide. In this paper we propose two important extensions of the generalized Prony method
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Scaling Limits of Jacobi Matrices and the Christoffel–Darboux Kernel Constr. Approx. (IF 1.531) Pub Date : 2020-02-20 Jonathan Breuer
We study scaling limits of deterministic Jacobi matrices, centered around a fixed point \(x_0\), and their connection to the scaling limits of the Christoffel–Darboux kernel at that point. We show that in the case when the orthogonal polynomials are bounded at \(x_0\), a subsequential limit always exists and can be expressed as a canonical system. We further show that under weak conditions on the associated
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Rhodonea Curves as Sampling Trajectories for Spectral Interpolation on the Unit Disk Constr. Approx. (IF 1.531) Pub Date : 2020-01-27 Wolfgang Erb
Rhodonea curves are classical planar curves in the unit disk with the characteristic shape of a rose. In this work, we use these rose curves as sampling trajectories to create novel nodes for spectral interpolation on the disk. By generating the interpolation spaces with a parity-modified Chebyshev–Fourier basis, we will prove the unisolvence of the interpolation on the rhodonea nodes. Properties such
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Besov Spaces Induced by Doubling Weights Constr. Approx. (IF 1.531) Pub Date : 2020-01-22 Atte Reijonen
Let \(1\leqslant p<\infty \), \(0
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Five Point Energy Minimization: A Synopsis Constr. Approx. (IF 1.531) Pub Date : 2020-02-13 Richard Evan Schwartz
This paper is a condensation of my arXiv monograph entitled Schwartz “The phase transition in 5 point energy minimization”, 2016. arXiv:1610.03303, which contains a complete proof that there is a constant such that the triangular bi-pyramid is the minimizer, amongst all 5 point configurations on the sphere, with respect to the power law potential \(R_s(r)=\mathrm{sign}(s)/r^s\), if and only if . In
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Correction to: Lattice Algorithms for Multivariate $$L_\infty $$ L ∞ Approximation in the Worst-Case Setting Constr. Approx. (IF 1.531) Pub Date : 2020-01-21 Frances Y. Kuo, Grzegorz W. Wasilkowski, Henryk Woźniakowski
We correct the expression for the worst-case error derived in [Kuo, Wasilkowski, Woźniakowski, Construct. Approx. 30 (2009), 475–493] and explain that the main theorem of the paper holds with enlarged constants.
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Minimal Soft Lattice Theta Functions Constr. Approx. (IF 1.531) Pub Date : 2020-01-09 Laurent Bétermin
We study the minimality properties of a new type of “soft” theta functions. For a lattice \(L\subset {\mathbb {R}}^d\), an L-periodic distribution of mass \(\mu _L\), and another mass \(\nu _z\) centered at \(z\in {\mathbb {R}}^d\), we define, for all scaling parameters \(\alpha >0\), the translated lattice theta function \(\theta _{\mu _L+\nu _z}(\alpha )\) as the Gaussian interaction energy between
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Existence of Meromorphic Solutions of First-Order Difference Equations Constr. Approx. (IF 1.531) Pub Date : 2019-12-17 Risto Korhonen; Yueyang Zhang
It is shown that if$$\begin{aligned} f(z+1)^n=R(z,f), \end{aligned}$$where R(z, f) is rational in f with meromorphic coefficients and \(\deg _f(R(z,f))=n\), has an admissible meromorphic solution, then either f satisfies a difference linear or Riccati equation with meromorphic coefficients, or the equation above can be transformed into one in a list of ten equations with certain meromorphic or algebroid
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Manifold Approximation by Moving Least-Squares Projection (MMLS) Constr. Approx. (IF 1.531) Pub Date : 2019-12-16 Barak Sober, David Levin
In order to avoid the curse of dimensionality, frequently encountered in big data analysis, there has been vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a lower-dimensional manifold; thus the high dimensionality problem can be overcome
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Pointwise and Uniform Convergence of Fourier Extensions Constr. Approx. (IF 1.531) Pub Date : 2019-11-19 Marcus Webb; Vincent Coppé; Daan Huybrechs
Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have
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Intertwining Operators Associated with Dihedral Groups Constr. Approx. (IF 1.531) Pub Date : 2019-11-15 Yuan Xu
The Dunkl operators associated with a dihedral group are a pair of differential-difference operators that generate a commutative algebra acting on differentiable functions in \({\mathbb {R}}^2\). The intertwining operator intertwines between this algebra and the algebra of differential operators. The main result of this paper is an integral representation of the intertwining operator on a class of
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Korovkin-Type Results on Convergence of Sequences of Positive Linear Maps on Function Spaces Constr. Approx. (IF 1.531) Pub Date : 2019-11-05 Maliheh Hosseini, Juan J. Font
In this paper, we deal with the convergence of sequences of positive linear maps to a (not assumed to be linear) isometry on spaces of continuous functions. We obtain generalizations of known Korovkin-type results and provide several illustrative examples.
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Sharp Constants of Approximation Theory. II. Invariance Theorems and Certain Multivariate Inequalities of Different Metrics Constr. Approx. (IF 1.531) Pub Date : 2019-10-04 Michael I. Ganzburg
We prove invariance theorems for general inequalities of different metrics and apply them to limit relations between the sharp constants in the multivariate Markov–Bernstein–Nikolskii type inequalities with the polyharmonic operator for algebraic polynomials on the unit sphere and the unit ball in \({\mathbb {R}}^m\) and the corresponding constants for entire functions of spherical type on \({\mathbb
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Zeros of Faber Polynomials for Joukowski Airfoils Constr. Approx. (IF 1.531) Pub Date : 2019-10-01 N. Levenberg; F. Wielonsky
Let K be the closure of a bounded region in the complex plane with simply connected complement whose boundary is a piecewise analytic curve with at least one outward cusp. The asymptotics of zeros of Faber polynomials for K are not understood in this general setting. Joukowski airfoils provide a particular class of such sets. We determine the (unique) weak-* limit of the full sequence of normalized
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Further Strengthening of Rolle’s Theorem for Complex Polynomials Constr. Approx. (IF 1.531) Pub Date : 2019-09-24 Blagovest Sendov, Hristo Sendov
A domain \(\Theta _n\) is called a Rolle’s domain if every complex polynomial p of degree n, satisfying \(p(i)=p(-i)\), has at least one critical point in it. In this paper, we find the smallest possible Rolle’s domain made up of two closed disks that are symmetric with respect to the real and the imaginary axes. This is a strengthening of the main result in Sendov and Sendov (Proc Am Math Soc 146(8):3367–3380
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Analysis of Decimation on Finite Frames with Sigma-Delta Quantization Constr. Approx. (IF 1.531) Pub Date : 2019-09-12 Kung-Ching Lin
In analog-to-digital conversion, signal decimation has been proven to greatly improve the efficiency of data storage while maintaining high accuracy. When one couples signal decimation with the \(\Sigma \Delta \) quantization scheme, the reconstruction error decays exponentially with respect to the bit-rate. In this study, similar results are proved for finite unitarily generated frames. We introduce
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Moment Preserving Local Spline Projection Operators Constr. Approx. (IF 1.531) Pub Date : 2019-07-15 Martin Campos Pinto
This article describes an elementary construction of a dual basis for nonuniform B-splines that is local, \(L^\infty \)-stable, and reproduces polynomials of any prescribed degree. This allows one to define local projection operators with near-optimal approximation properties in any \(L^q\), \(1 \le q \le \infty \), and high order moment preserving properties. As the dual basis functions share the
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Conditionally Positive Definite Matrix Valued Kernels on Euclidean Spaces Constr. Approx. (IF 1.531) Pub Date : 2019-07-12 J. C. Guella; V. A. Menegatto
The purpose of this paper is to provide necessary and sufficient conditions on a continuous and matrix valued radial kernel on a Euclidean space in order that it be conditionally positive definite of a fixed order. Except for the one dimensional Euclidean space, the strict conditional positive definiteness of the kernel is fully characterized.
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Sampling Almost Periodic and Related Functions Constr. Approx. (IF 1.531) Pub Date : 2019-07-11 Stefano Ferri, Jorge Galindo, Camilo Gómez
We consider certain finite sets of circle-valued functions defined on intervals of real numbers and estimate how large the intervals must be for the values of these functions to be uniformly distributed in an approximate way. This is used to establish some general conditions under which a random construction introduced by Katznelson for the integers yields sets that are dense in the Bohr group. We
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A Discrete Realization of the Higher Rank Racah Algebra Constr. Approx. (IF 1.531) Pub Date : 2019-07-11 Hendrik De Bie; Wouter van de Vijver
In previous work, a higher rank generalization R(n) of the Racah algebra was defined abstractly. The special case of rank one encodes the bispectrality of the univariate Racah polynomials and is known to admit an explicit realization in terms of the operators associated with these polynomials. Starting from the Dunkl model for which we have an action by R(n) on the Dunkl-harmonics, we show that connection
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Quadrature Points via Heat Kernel Repulsion Constr. Approx. (IF 1.531) Pub Date : 2019-07-11 Jianfeng Lu; Matthias Sachs; Stefan Steinerberger
We discuss the classical problem of how to pick N weighted points on a d-dimensional manifold so as to obtain a reasonable quadrature rule$$\begin{aligned} \frac{1}{|M|}\int _{M}{f(x) \mathrm{d}x} \simeq \sum _{n=1}^{N}{a_i f(x_i)}. \end{aligned}$$This problem, naturally, has a long history; the purpose of our paper is to propose selecting points and weights so as to minimize the energy functional$$\begin{aligned}
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Convergence and Normal Continuity Analysis of Nonstationary Subdivision Schemes Near Extraordinary Vertices and Faces Constr. Approx. (IF 1.531) Pub Date : 2019-07-05 Costanza Conti; Marco Donatelli; Lucia Romani; Paola Novara
Convergence and normal continuity analysis of a bivariate nonstationary (level-dependent) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, in this paper we derive new sufficient conditions for establishing convergence and normal continuity of any rotationally symmetric, nonstationary
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Sharp Remez Inequality Constr. Approx. (IF 1.531) Pub Date : 2019-07-01 S. Tikhonov, P. Yuditskii
Let an algebraic polynomial \(P_n(\zeta )\) of degree n be such that \(|P_n(\zeta )|\leqslant 1\) for \(\zeta \in E\subset \mathbb {T}\) and \(|E|\geqslant 2\pi -s\). We prove the sharp Remez inequality $$\begin{aligned} \sup _{\zeta \in \mathbb {T}}|P_n(\zeta )|\leqslant {\mathfrak {T}}_{n}\left( \sec \frac{s}{4}\right) , \end{aligned}$$ where \({\mathfrak {T}}_{n}\) is the Chebyshev polynomial of
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Bounds for $$L_p$$Lp -Discrepancies of Point Distributions in Compact Metric Measure Spaces Constr. Approx. (IF 1.531) Pub Date : 2019-07-01 M. M. Skriganov
Upper bounds for the \(L_p\)-discrepancies of point distributions in compact metric measure spaces are proved for all exponents \(0
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Rigidity of the Hamburger and Stieltjes Moment Sequences Constr. Approx. (IF 1.531) Pub Date : 2019-06-25 Alexander Dyachenko
This paper aims at finding conditions on a Hamburger or Stieltjes moment sequence, under which the change of at most a finite number of its entries produces another sequence of the same type. It turns out that a moment sequence allows all small enough variations of this kind precisely when it is indeterminate. We also show that a determinate moment sequence has the finite index of determinacy if and
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Fejér Polynomials and Control of Nonlinear Discrete Systems Constr. Approx. (IF 1.531) Pub Date : 2019-06-25 D. Dmitrishin; P. Hagelstein; A. Khamitova; A. Korenovskyi; A. Stokolos
We consider optimization problems associated with a delayed feedback control (DFC) mechanism for stabilizing cycles of one-dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing T-cycles of a differentiable function \(f: \mathbb {R}\rightarrow \mathbb {R}\) of the form$$\begin{aligned} x(k+1) = f(x(k)) + u(k), \end{aligned}$$where$$\begin{aligned} u(k)
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Weighted Norm Inequalities for Fractional Bergman Operators Constr. Approx. (IF 1.531) Pub Date : 2019-06-25 Benoît F. Sehba
We prove in this paper one weight norm inequalities for some positive Bergman-type operators.
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Double Affine Hecke Algebra of Rank 1 and Orthogonal Polynomials on the Unit Circle Constr. Approx. (IF 1.531) Pub Date : 2019-06-20 Satoshi Tsujimoto; Luc Vinet; Alexei Zhedanov
An infinite-dimensional representation of the double affine Hecke algebra of rank 1 and type \((C_1^{\vee },C_1)\) in which all generators are tridiagonal is presented. This representation naturally leads to two systems of polynomials that are orthogonal on the unit circle. These polynomials can be considered as circle analogs of the Askey–Wilson polynomials. The corresponding polynomials orthogonal
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Compressive Hermite Interpolation: Sparse, High-Dimensional Approximation from Gradient-Augmented Measurements Constr. Approx. (IF 1.531) Pub Date : 2019-06-11 Ben Adcock; Yi Sui
We consider the sparse polynomial approximation of a multivariate function on a tensor product domain from samples of both the function and its gradient. When only function samples are prescribed, weighted \(\ell ^1\) minimization has recently been shown to be an effective procedure for computing such approximations. We extend this work to the gradient-augmented case. Our main results show that for
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Rational Solutions of the Painlevé-III Equation: Large Parameter Asymptotics Constr. Approx. (IF 1.531) Pub Date : 2019-05-29 Thomas Bothner; Peter D. Miller
The Painlevé-III equation with parameters \(\Theta _0=n+m\) and \(\Theta _\infty =m-n+1\) has a unique rational solution \(u(x)=u_n(x;m)\) with \(u_n(\infty ;m)=1\) whenever \(n\in \mathbb {Z}\). Using a Riemann–Hilbert representation proposed in Bothner et al. (Stud Appl Math 141:626–679, 2018), we study the asymptotic behavior of \(u_n(x;m)\) in the limit \(n\rightarrow +\infty \) with \(m\in \mathbb
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On Accumulated Cohen’s Class Distributions and Mixed-State Localization Operators Constr. Approx. (IF 1.531) Pub Date : 2019-05-28 Franz Luef; Eirik Skrettingland
Recently we introduced mixed-state localization operators associated with a density operator and a (compact) domain in phase space. We continue the investigations of their eigenvalues and eigenvectors. Our main focus is the definition of a time-frequency distribution that is based on the Cohen class distribution associated with the density operator and the eigenvectors of the mixed-state localization
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Asymptotic Greediness of the Haar System in the Spaces $$L_p[0,1]$$Lp[0,1] , $$1 Constr. Approx. (IF 1.531) Pub Date : 2019-05-28 Fernando Albiac; José L. Ansorena; Pablo M. Berná
Our aim in this paper is to attain a sharp asymptotic estimate for the greedy constant \(C_{g}[\mathcal {H}^{(p)},L_p]\) of the (normalized) Haar system \(\mathcal {H}^{(p)}\) in \(L_{p}[0,1]\) for \(1
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Sharp Approximations for the Ramanujan Constant Constr. Approx. (IF 1.531) Pub Date : 2019-05-17 Song-Liang Qiu; Xiao-Yan Ma; Ti-Ren Huang
In this paper, the authors present sharp approximations in terms of sine function and polynomials for the so-called Ramanujan constant (or the Ramanujan R-function) R(a), by showing some monotonicity, concavity and convexity properties of certain combinations defined in terms of R(a), \(\sin (\pi a)\) and polynomials. Some properties of the Riemann zeta function and its related special sums are presented
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Learning General Sparse Additive Models from Point Queries in High Dimensions Constr. Approx. (IF 1.531) Pub Date : 2019-05-03 Hemant Tyagi; Jan Vybiral
We consider the problem of learning a d-variate function f defined on the cube \([-1,1]^d\subset \mathbb {R}^d\), where the algorithm is assumed to have black box access to samples of f within this domain. Let \({\mathcal {S}}_r \subset {[d] \atopwithdelims ()r}; r=1,\dots ,r_0\) be sets consisting of unknown r-wise interactions amongst the coordinate variables. We then focus on the setting where f
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Rational Extension of the Newton Diagram for the Positivity of $${}_1F_2$$1F2 Hypergeometric Functions and Askey–Szegö Problem Constr. Approx. (IF 1.531) Pub Date : 2019-05-03 Yong-Kum Cho; Seok-Young Chung; Hera Yun
We present a rational extension of the Newton diagram for the positivity of \({}_1F_2\) generalized hypergeometric functions. As an application, we give upper and lower bounds for the transcendental roots \(\beta (\alpha )\) of$$\begin{aligned} \int _0^{j_{\alpha , 2}} t^{-\beta } J_\alpha (t) \hbox {d}t = 0\qquad (-\,1<\alpha \le 1/2), \end{aligned}$$where \(j_{\alpha , 2}\) denotes the second positive
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Morrey Sequence Spaces: Pitt’s Theorem and Compact Embeddings Constr. Approx. (IF 1.531) Pub Date : 2019-03-28 Dorothee D. Haroske; Leszek Skrzypczak
Morrey (function) spaces and, in particular, smoothness spaces of Besov–Morrey or Triebel–Lizorkin–Morrey type have enjoyed a lot of interest recently. Here we turn our attention to Morrey sequence spaces \(m_{u,p}=m_{u,p}(\mathbb {Z}^d)\), \(0
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Menshov Type Correction Theorems for Sequences of Compact Operators Constr. Approx. (IF 1.531) Pub Date : 2019-03-28 Grigori A. Karagulyan
We prove Menshov type “correction” theorems for sequences of compact operators, recovering several results of Fourier series in trigonometric and Walsh systems. The paper clarifies the main ingredient that is important in the study of such “correction” theorems. That is the weak-\(L^1\) estimate for the maximal Fourier sums of indicator functions of some specific sets.
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