Best approximation in Hardy spaces and by polynomials, with norm constraints
type: Item
instance of: scholarly article
title: Best approximation in Hardy spaces and by polynomials, with norm constraints
zbMATH Open document ID: 1275.41032
author: J. Leblond
author: Elodie Pozzi
author: Jonathan R. Partington
published in: Integral Equations and Operator Theory
publication date: 2013-05-06T00:00:00Z
full work available at URL: RR-8098.pdf
review text: Let \(G\) be either the disk or the annulus. Let \(I\) be a subset of the boundary of \(G\) so that both \(I\) and its complement \(J\) in the boundary of \(G\) have positive Lebesgue measure. Let \(f\) be a Lebesgue integrable function on \(I\) and let \(c\) and \(M > 0\) be given numbers. The problem of this study is to find a solution \(g\) so that the \(L_p\)-norm over \(I\) of \(f-g_c\) is equal \[ \min_g\{\| f-g_c\|_{L_p(I)} , g \in H^p(G) ; \|g -c\|_{L_p(J )}\leq M \}; \] where \( H^p(G) \) represents the Hardy space associated with \(p\) and \(G\). Truncated versions of the problem are analyzed with solutions chosen from finite dimensional spaces of polynomials or rational functions. The solutions are presented in terms of truncated Toeplitz operators. The results are illustrated with numerical examples. The results are related to inverse problems for PDEs.
Mathematics Subject Classification ID: 41A40
Mathematics Subject Classification ID: 41A45
zbMATH DE Number: 6160907
zbMATH Keywords: subsets of boundary of the disk or annulus
zbMATH Keywords: \(L_p\)-norm
zbMATH Keywords: rational approximates
reviewed by: Daniel Wulbert
MaRDI profile type: Publication
OpenAlex ID: W1990883437
Wikidata QID: Q56319162
cites work: Toeplitz operators in multiply connected regions
cites work: Q4029556
cites work: Traces of Hardy functions and reproducing kernel Hilbert spaces
cites work: On the existence of eigenvalues of Toeplitz operators on planar regions
cites work: Bounded extremal and Cauchy-Laplace problems on the sphere and shell
cites work: Asymptotic estimates for interpolation and constrained approximation in \(H^2\) by diagonalization of Toeplitz operators
cites work: Hardy approximation to \(L^p\) functions on subsets of the circle with \(1\leq p<\infty\)
cites work: Hardy approximation to \(L^ \infty\) functions on subsets of the circle
cites work: Robust identification from band-limited data
cites work: Q3711012
cites work: Introduction to large truncated Toeplitz matrices
cites work: The spectral theory of Toeplitz operators applied to approximation problems in Hilbert spaces
cites work: Q4950748
cites work: Constrained approximation and invariant subspaces
cites work: Approximation in reflexive Banach spaces and applications to the invariant subspace problem
cites work: Q3949869
cites work: Q5618011
cites work: Analytic extensions and Cauchy-type inverse problems on annular domains: stability results
cites work: Bounded extremal problems in Hardy spaces for the conjugate Beltrami equation in simply-connected domains
cites work: Robust numerical algorithms based on analytic approximation for the solution of inverse problems in annular domains
cites work: Analytic approximation with real constraints, with applications to inverse diffusion problems
cites work: Slepian functions on the sphere, generalized Gaussian quadrature rule
cites work: Q4537721
cites work: Q4000499
cites work: Q3838090
cites work: The 𝐻^{𝑝} spaces of an annulus
cites work: Algebraic properties of truncated Toeplitz operators
cites work: Spatiospectral Concentration on a Sphere
cites work: Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty-V: The Discrete Case
cites work: Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - I
DOI: 10.1007/S00020-013-2036-6
description: scientific article; zbMATH DE number 6160907