Continuous and discrete tight frames of orthogonal polynomials for a radially symmetric weight
type: Item
instance of: scholarly article
title: Continuous and discrete tight frames of orthogonal polynomials for a radially symmetric weight
zbMATH Open document ID: 1180.33011
published in: Constructive Approximation
publication date: 2009-08-31T00:00:00Z
review text: The author constructs orthogonal-type expansions with the simplest possible form for the space \(\mathcal{P}_n\) of orthogonal polynomials of degree \(n\) for a general radially symmetric weight function on \(\mathbb{R}^d\). The main result of this paper is: For any \(d \geq 2\), the explicit construction of a zonal polynomial \(p \in \mathcal{P}_n\) having the following property. Each \(f \in \mathcal{P}_n\) can be reconstructed as a sum of its projections onto the orbit of \(p\) under \(SO(d)\) (symmetries of the weight), and hence of its projections onto the zonal polynomials \(p_{\xi}\) obtained from \(p\) by moving its pole to the set \( \{\xi \in \mathbb{R}^d: |\xi|=1\}\). The approach followed here is first to prove the continuous versions and then to deduce the discrete decompositions. Finally, an application of the continuous tight frame decomposition is presented to obtain a simple closed form for the reproducing kernel of \(\mathcal{P}_n\).
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33D50
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 5600301
zbMATH Keywords: multivariate orthogonal polynomials
zbMATH Keywords: Jacobi polynomials
zbMATH Keywords: Gegenbauer polynomials
zbMATH Keywords: ultraspherical polynomials
zbMATH Keywords: Legendre polynomials
zbMATH Keywords: Hermite polynomials
zbMATH Keywords: Laguerre polynomials
zbMATH Keywords: harmonic functions
zbMATH Keywords: spherical harmonics
zbMATH Keywords: zonal harmonics
zbMATH Keywords: quadrature for trigonometric polynomials
zbMATH Keywords: cubature on the sphere
zbMATH Keywords: representation theory
zbMATH Keywords: tight frames
author: Shayne Waldron
MaRDI profile type: Publication
full work available at URL: s00365-008-9021-3
OpenAlex ID: W1973726364
cites work: Q4023465
cites work: A new form of the spherical expansion of zonal functions and Fourier transforms of \(\text{SO}(d)\)-finite functions
cites work: An introduction to frames and Riesz bases
cites work: Cubature formulae and orthogonal polynomials
cites work: Orthogonal polynomials of several variables
cites work: Deux cours d'analyse harmonique. École d'Été d'Analyse Harmonique de Tunis, 1984
cites work: Spherical designs and finite group representations (some results of E. Bannai).
cites work: Q3140236
cites work: Optimal reconstruction of a function from its projections
cites work: Approximation by Ridge Functions and Neural Networks
cites work: Q3884590
cites work: Definitions for spherical designs
cites work: Q5640160
cites work: Q4158394
cites work: Tight frames and their symmetries
cites work: Hermite polynomials on the plane
cites work: Orthogonal polynomials on the disc
cites work: Summability of Fourier orthogonal series for Jacobi weight on a ball in ℝ^{𝕕}
cites work: Representation of reproducing kernels and the Lebesgue constants on the ball
cites work: Q5295584
cites work: Reconstruction from Radon projections and orthogonal expansion on a ball
DOI: 10.1007/S00365-008-9021-3
description: scientific article; zbMATH DE number 5600301