On a filter for exponentially localized kernels based on Jacobi polynomials
type: Item
instance of: scholarly article
title: On a filter for exponentially localized kernels based on Jacobi polynomials
zbMATH Open document ID: 1203.42038
author: Jürgen Prestin
author: Frank Filbir
author: H. N. Mhaskar
published in: Journal of Approximation Theory
publication date: 2009-12-03T00:00:00Z
review text: The problem of detection of singularities of a function from spectral data and the closely related problem of spectral approximation of piecewise smooth functions arise in many applications. The locations of singularities are found using an appropriate filter, and then pseudo-spectral methods are used on the maximum intervals of smoothness to compute the approximation on these intervals. A filter is a bi-infinite matrix \(H,\) and the corresponding mollifier is given by \[ \Phi_n(H, \theta)=\sum_{k\in \mathbb Z}H_{k,n}\exp(ik\theta), \quad \sigma_n(H, f, \theta)=\frac{1}{2\pi}\int_{-\pi}^\pi f(\phi) \Phi_n(H, \theta -\phi)d\phi , \theta\in {\mathbb{R}} . \] The purpose of this paper is to demonstrate the construction of a filter in the case of Jacobi polynomials. Let \(\alpha,\beta\geq-\frac{1} {2},\) and for \(k=0,1,\dots, p_{k}^{(\alpha,\beta)}\) denote the orthonormalized Jacobi polynomial of degree \(k,\) so that \[ \int_{-1}^1 p_{k}^{(\alpha,\beta)}(x) p_{m}^{(\alpha,\beta)}(x) w_{\alpha,\beta}(x)dx=\delta_{k,m}, \] where \(w_{\alpha,\beta}(x)=(1-x)^\alpha (1+x)^\beta, -1\leq x\leq 1.\) The authors introduce a translation operator \(\tau_{y}\) and use it to define a convolution operator by \[ \left( f\ast g\right) (x):=\int_{-1}^{1}f(y)\tau_{y}g(x)w_{\alpha,\beta }(y)dy,\;\;\;f,g\in L^{1}\left( \alpha,\beta\right) \] so that \[ \widehat{( f\ast g)}(k)=\left\{ p_{k}^{\left( \alpha ,\beta\right) }\left( 1)\right) \right\} ^{-1}\hat{f}(k)\hat{g}(k), \] where \[ \hat{f}(k)=\int_{-1}^1 f(x) p_{k}^{(\alpha,\beta)}(x)w_{\alpha,\beta}(x)dx. \] The authors construct a kernel localized at \(x=1\) and then extend the result by using translation theorems they developed and obtain a matrix \(H\) satisfying \[ \begin{aligned} \left| \sum_{k=0}^{\infty}H_{k,n}p_{k}^{(\alpha,b)}\left( \cos \theta\right) p_{k}^{(\alpha,\beta)}(\cos\varphi)\right| & \leq c_{1}n^{2\max(\alpha,\beta)+2}\exp(-cn\left( \theta-\varphi\right) ^{2}),\\ & \theta,\varphi\in[0,\pi],n=1,2,\dots \end{aligne…
reviewed by: Ahmed I. Zayed
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 42C30
zbMATH DE Number: 5641735
zbMATH Keywords: Jacobi polynomials
zbMATH Keywords: localized filters and mollifiers
zbMATH Keywords: spectral approximations
zbMATH Keywords: Riesz basis
describes a project that uses: OPQ
MaRDI profile type: Publication
full work available at URL: j.jat.2009.01.004
OpenAlex ID: W1965277554
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DOI: 10.1016/J.JAT.2009.01.004
description: scientific article; zbMATH DE number 5641735