Exponentially accurate approximations to periodic Lipschitz functions based on Fourier series partial sums
type: Item
instance of: scholarly article
title: Exponentially accurate approximations to periodic Lipschitz functions based on Fourier series partial sums
zbMATH Open document ID: 0934.65146
DOI: 10.1023/A:1023289301743
author: James F. Geer
author: Nana Saheb Banerjee
published in: Journal of Scientific Computing
publication date: 1999-11-24T00:00:00Z
review text: The purpose of this interesting paper is to present an accurate and robust technique to estimate the locations of discontinuities and the associated jumps in a piecewise smooth, \(2\pi\)-periodic function \(f\), if the first \(2N+ 1\) Fourier coefficients of \(f\) are known. The method consists of two steps: 1. Find initial estimates of the locations and magnitudes of the discontinuities in \(f\) by certain properties of Gibbs phenomenon. 2. Refine these estimates by fitting the asymptotic form of the Fourier coefficients to the given coefficients using a least squares method. Then this procedure, the so-called least squares parameter estimation method, is extended to find discontinuities in \(f^{(k)}\) \((k=1,\dots, M)\). Some aspects of the robustness of this method are discussed. Further, this method is used to reconstruct the original function \(f\), as well as its first few derivatives, with exponential accuracy, if the Fourier coefficients of \(f\) are known. The technique is illustrated with several examples.
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 65T40
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 42A10
zbMATH DE Number: 1346134
zbMATH Keywords: reconstruction of periodic
zbMATH Keywords: piecewise smooth function
zbMATH Keywords: exponentially accurate approximation
zbMATH Keywords: Fourier sum
zbMATH Keywords: numerical examples
zbMATH Keywords: locations of discontinuities
zbMATH Keywords: Gibbs phenomenon
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: least-squares method
zbMATH Keywords: parameter estimation
MaRDI profile type: Publication
full work available at URL: a:1023289301743
OpenAlex ID: W37489506
description: scientific article; zbMATH DE number 1346134