Some extremal problems for polynomials in integral metrics
type: Item
instance of: scholarly article
title: Some extremal problems for polynomials in integral metrics
zbMATH Open document ID: 0855.42001
DOI: 10.1007/BF02304899
author: A. V. Moskovskij
published in: Mathematical Notes
publication date: 1997-02-03T00:00:00Z
review text: The author studies the behavior of the functions: \[ \Phi_{pq} (n, \eta) : = \sup \bigl\{ |p_{n - 1} |_{q [- \eta, \eta]} \mid p_{n - 1} \in P_{n - 1},\;|p_{n - 1} |_{p [- 1, 1]} \leq 1 \bigr\} \] and \[ \Psi_{pq} (n,a) : = \sup \bigl\{ |t_{n - 1} |_{q [- \pi, \pi]} \mid t_{n - 1} \in T_{n - 1},\;|t_{n - 1} |_{p [- a, a]} \leq 1 \bigr\}, \] where \(\eta \geq 1\), \(0 < a \leq \pi\), \(1 \leq p\), \(q \leq \infty\) and \[ |f |_{p [a,b]} : = \left( \int^b_a \bigl |f(x) \bigr |^p dx \right)^{1/p}, \quad |f |_{\infty [a,b]} : = \max_{[a,b]} \bigl |f(x) \bigr |, \] furthermore, \(P_n\) and \(T_n\) denote the set of \(n\)th-order algebraic and trigonometric polynomials. Among others it is proved that for \(1 \leq p,q \leq \infty\), the following relation is valid uniformly in \(n \in \mathbb{N}\) and \(\eta \geq 1\): \[ \Phi_{pq} (n, \eta) \asymp \max \left\{ 1, {n^{2 (1/p - 1/q)} \bigl( g (\eta) \bigr)^{n - 1 + 1/p + 1/2q} \over \bigl( n \sqrt {\eta - 1} + 1 \bigr )^{2/p - 1/q}} \right\}, \] where \(g (\eta) : = \eta + \sqrt {\eta^2 - 1}\).
Mathematics Subject Classification ID: 42A05
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 921650
zbMATH Keywords: algebraic polynomials
zbMATH Keywords: extremal problems
zbMATH Keywords: integral metrics
zbMATH Keywords: trigonometric polynomials
reviewed by: László Leindler
MaRDI profile type: Publication
cites work: Certain problems of closest approximation
description: scientific article; zbMATH DE number 921650