Jackson's theorem in the space \(L_{2}(\mathbb R^{d})\) with power weight
type: Item
instance of: scholarly article
title: Jackson's theorem in the space \(L_{2}(\mathbb R^{d})\) with power weight
zbMATH Open document ID: 1231.41010
DOI: 10.1134/S0001434610070126
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: Suppose that \(d\in\mathbb{N}\), \(\mathbb{R}^d\) is \(d\)-dimensional Euclidean space with inner product \((x,y)=\sum_{i=1}^dx_iy_i\) and modulus \(|x|=\sqrt{(x,x)}\) for all \(x,\,y\in\mathbb{R}^d\), \(\lambda=(\lambda_1,\ldots,\lambda_d)\in \mathbb{R}^d\), \(\lambda_i\in [-1/2,\infty)\), \(i\in\{1,\ldots,d\}\), \(\sigma(\lambda)=\lambda_1+\cdots+\lambda_d\), \[ v_{\lambda}(x)=\frac1{2^{\sigma(\lambda)+d}\prod_{i=1}^d\Gamma(\lambda_i+1)} \prod_{i=1}^d|x_i|^{2\lambda_i+1} \] for all \(x\in \mathbb{R}^d\) is the power weight in \(\mathbb{R}^d\), \(d\mu_{\lambda}(x)=v_{\lambda}(x)dx\), and \(L_{2,\lambda}(\mathbb{R}^d)\) is the space of complex Lebesgue measurable functions \(f\) on \(\mathbb{R}^d\) with finite norm \[ \|f\|_{2,\lambda}=\left[\int_{\mathbb{R}^d}|f(x)|^2 \,d\mu_{\lambda}(x)\right]^{1/2}<\infty. \] For \(\lambda_i=-1/2\), \(i\in\{1,\ldots,d\}\), \(L_{2,\lambda}(\mathbb{R}^d)\) is simply denoted by \(L_2(\mathbb{R}^d)\). Assume that \(E_R(f)_2\) is the value of the best approximation of a function \(f\) by entire functions of exponential spherical type \(R\in (0,\infty)\) in the space \(L_2(\mathbb{R}^d)\), \[ \omega(\delta,f)_2=\sup_{|t|\leq\delta}\|f(x+t)-f(x)\|_{L_2(\mathbb{R}^d)} \] is its modulus of continuity, and \(q_{\alpha}\) is the least positive zero of the Bessel function \(J_{\alpha}\) of order \(\alpha\in[-1/2,\infty)\). The sharp Jackson inequality in the space \(L_2(\mathbb{R}^d)\) \[ E_R(f)_2\leq\frac1{\sqrt{2}}\omega\left(\frac{2q_{d/2-1}}R,f\right)_2 \] is well known. Let \(R\in (0,\infty)\), \(B_R=\{x\in\mathbb{R}^d\,:\,|x|\leq R\}\), \(E^R_{2,\lambda}\) be the set of entire functions \(g\in L_{2,\lambda}(\mathbb{R}^d)\) of exponential spherical type \(R\) for which the support \({\displaystyle\text{supp}}\, \widehat{g}\subset B_R\), and \(E_R(f)_{2,\lambda} =\inf\{\|f-g\|_{2,\lambda}\,:\,g\in E^R_{2,\lambda}\}\) be the value of the best approximation of the function \(f\in L_{2,\lambda}(\mathbb{R}^d)\) and \[ \omega_1(\delta,f)_{2,\…
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 41A80
zbMATH DE Number: 5980656
zbMATH Keywords: Jackson's theorem
zbMATH Keywords: Euclidean space
zbMATH Keywords: \(L_2(\mathbb{R}^d)\)
zbMATH Keywords: Bessel function
zbMATH Keywords: Dunkl operator
zbMATH Keywords: Dunkl's Laplacian
zbMATH Keywords: Parseval's equality
author: Alexey Ivanov
author: Valeriĭ I. Ivanov
reviewed by: Dachun Yang
MaRDI profile type: Publication
cites work: The Dunkl transform
cites work: Funk-Hecke Formula for Orthogonal Polynomials on Spheres and on Balls
description: scientific article; zbMATH DE number 5980656