The rate of convergence of Fourier expansions in the plane: a geometric viewpoint
type: Item
instance of: scholarly article
title: The rate of convergence of Fourier expansions in the plane: a geometric viewpoint
zbMATH Open document ID: 1047.42008
author: Luca Brandolini
author: Leonardo Colzani
author: Giancarlo Travaglini
author: Alex Iosevich
published in: Mathematische Zeitschrift
publication date: 2004-06-17T00:00:00Z
review text: Let \(f= g\chi_B\), where \(g\) is a smooth function on \(\mathbb{R}^2\) and \(B\) is a bounded planar domain with smooth boundary, and let \[ S_\lambda f(x)= \int_{\lambda A}\widehat f(\xi)\exp(2\pi i\xi\cdot x)\,d\xi, \] where \(A\) is a bounded planar domain with smooth boundary and strictly starshaped with respect to the origin and \[ \widehat f(\xi):= \int_{\mathbb{R}^2} f(y)\exp(-2\pi i\xi\cdot y)\,dy. \] One of the main results in the paper is the following Theorem 1: There exist constants \(c= c(x,A,B,g)\) and \(\ell= \ell(x,A,B)\) such that \[ \begin{aligned}| S_\lambda f(x)|\leq c\lambda^{-1/2}\quad &\text{if }x\in\overline B,\\ | S_\lambda f(x)- g(x)|\leq c\lambda^{-1/2}\quad &\text{if }x\in B,\\ | S_\lambda f(x)- \ell g(x)|\leq c\lambda^{-1/2}\quad &\text{if }x\in\partial B.\end{aligned} \] We note that in the special case when \(A\) is a disc, it is well known that \(\ell(x,A,B)= 1/2\).
Mathematics Subject Classification ID: 42B08
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 2075750
zbMATH Keywords: Fourier expansion
zbMATH Keywords: Fourier transform
zbMATH Keywords: convergence
reviewed by: Ferenc Móricz
MaRDI profile type: Publication
full work available at URL: s002090100375
OpenAlex ID: W2023440678
DOI: 10.1007/S002090100375
description: scientific article; zbMATH DE number 2075750