Optimal quadrature evaluation of integrals of rapidly oscillating functions in Lipschitz interpolation class in case of strong oscillation
type: Item
instance of: scholarly article
title: Optimal quadrature evaluation of integrals of rapidly oscillating functions in Lipschitz interpolation class in case of strong oscillation
zbMATH Open document ID: 1221.65349
author: V. K. Zadiraka
author: L. V. Luts
author: S. S. Mel'nikova
published in: Cybernetics and Systems Analysis
publication date: 2010-09-24T00:00:00Z
review text: The authors consider the following integrals \[ I_{1}(\omega)=\int_{0}^{1}f(x)\sin\omega x dx,\,\,\,I_{2}(\omega)=\int_{0}^{1}f(x)\cos\omega x dx \tag{1} \] where \(f\in F\) (\(F\) is a given class of functions), \(\omega\) an arbitrary real number, \(|\omega|\geq 2\pi\), and the information about \(f\) is available at no more than \(N\) points. They find optimal estimates and derive accuracy-optimal quadrature formulas to evaluate the Fourier transform of a compact function on the assumption that \(f\in C_{L,N}\), where \(C_{L,N}\) is an interpolation class of functions that satisfy the Lipschitz property (\(|f(x_{1})-f(x_{2})| \leq L|x_{1}-x_{2}|\), \(x_{1},x_{2} \in [0,1] \)) and are defined by \(2N\) fixed values. The case of strong oscillation of the integrand is considered. The optimality is substantiated based on the boundary function method, namely, constructing the Chebyshev center and Chebyshev radius in the uncertainty domain of the problem solution.
reviewed by: Ana Maria Acu
Mathematics Subject Classification ID: 65T40
Mathematics Subject Classification ID: 42A16
zbMATH DE Number: 5790971
zbMATH Keywords: rapidly oscillating functions
zbMATH Keywords: accuracy-optimal algorithm
zbMATH Keywords: quadrature formulas
zbMATH Keywords: Fourier transform
zbMATH Keywords: boundary function method
zbMATH Keywords: Chebyshev center
zbMATH Keywords: Chebyshev radius
zbMATH Keywords: uncertainty domain
MaRDI profile type: Publication
full work available at URL: s10559-010-9203-4
OpenAlex ID: W2067870333
cites work: Q5665610
cites work: Optimal quadrature and cubature formulas for computing Fourier transform of finite functions of one class. Case of strong oscillation
cites work: Q3326801
DOI: 10.1007/S10559-010-9203-4
description: scientific article; zbMATH DE number 5790971