Orthogonal polynomials in \(L^ 1\)-approximation
type: Item
instance of: scholarly article
title: Orthogonal polynomials in \(L^ 1\)-approximation
zbMATH Open document ID: 0642.42023
DOI: 10.1016/0021-9045(88)90041-X
author: Franz Peherstorfer
published in: Journal of Approximation Theory
publication date: 1988-01-01T00:00:00Z
review text: Suppose that \(b_ 1,b_ 2,...,b_ n\) are given real numbers. The author considers the problem of determining a function h on \([0,\pi]\) with \(\ell\) changes of sign on \([0,\pi]\) (\(\ell \geq n)\), such that \(\int^{\pi}_{0}\sin (k\phi)h(\phi)d\phi =b_ k\) \((k=1,...,n)\). It is shown that the points at which the function h changes sign depend (under appropriate conditions for the numbers \(\{b_ k\}\) in a certain manner on some orthogonal polynomials. As an application the author considers Posse's problem: for which a and b \((1<a<b)\) is there a polynomial \(\tilde q_ n(x)=x^ n+..\). such that for every monic polynomial \(q_ n\) of degree n \(\int^{1}_{0} |\tilde q_ n| + (-1)^ n\int^{b}_{a}\tilde q_ n\leq \int^{1}_{a} | q_ n| + (- 1)^ n\int^{b}_{a}q_ n;\) furthermore find a minimizing polynomial when the conditions are satisfied. Posse solved this problem with the help of elliptic functions; the present author expresses the minimizing polynomials in terms of orthogonal polynomials on two intervals. Further applications are given in \(L_ 1\)-approximation on several intervals, in particular it is shown that there is a close connection between \(L_ 1\)- ,L\({}_ 2\)- and \(L_{\infty}\)-approximation with respect to a suitable weight on two intervals.
reviewed by: Walter Van Assche
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 4046355
zbMATH Keywords: Posse's problem
zbMATH Keywords: minimizing polynomials
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: \(L_ 1\)-approximation
zbMATH Keywords: weight
MaRDI profile type: Publication
cites work: Q4752839
cites work: Q5760170
cites work: Q5816262
cites work: Q5727877
cites work: Padé-type approximation and general orthogonal polynomials
cites work: Q5771410
cites work: Q4196693
cites work: Q4134389
cites work: Approximation in the Metric of L 1 (X,μ)
cites work: Simplified Treatment of Integrals of Cauchy Type, the Hilbert Problem and Singular Integral Equations. Appendix: Poincaré-Bertrand Formula
cites work: Q4150643
cites work: Trigonometric polynomial approximation in the L1-norm
cites work: On a sufficient condition in \(L^ 1-\)approximation
cites work: Q3341338
cites work: Best Approximation in an Asymmetrically Weighted L1 Measure
cites work: The Zeros of Certain Jacobi Polynomials
description: scientific article; zbMATH DE number 4046355