Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes.
type: Item
instance of: scholarly article
title: Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes.
zbMATH Open document ID: 1038.42002
author: Tamás Erdélyi
author: Peter Borwein
published in: Journal of Approximation Theory
publication date: 2004-01-14T00:00:00Z
review text: The authors give shorter and more direct proofs of certain results by J. E. Littlewood published in the 60's. These proofs also lead to explicit constants. The following theorem proved by them is typical: If \(p\) is a trigonometric polynomial of degree at most \(n\) with real coefficients that satisfies the conditions \[ \Biggl(\int_{\mathbb{T}} p^2(t)\,dt\Biggr)^{1/2}\leq An^{1/2}\quad\text{and}\quad \Biggl(\int_{\mathbb{T}}| p'(t)|^2\,dt\Biggr)^{1/2}\geq Bn^{3/2}, \] then \[ \Biggl({1\over 2\pi} \int_{\mathbb{T}} p^4(t)\,dt\Biggr)^{1/4}\geq \Biggl(1+{1\over 111} \Biggl({B\over A}\Biggr)^{1/2}\Biggr) \Biggl({1\over 2\pi} \int_{\mathbb{T}} p^2(t)\,dt\Biggr)^{1/2}. \] Conjugate reciprocal, unimodular polynomials on the unit circle of the complex plane are also considered.
Mathematics Subject Classification ID: 42A05
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 2026879
zbMATH Keywords: trigonometric polynomials
zbMATH Keywords: merit factor
zbMATH Keywords: unimodular trigonometric polynomials
zbMATH Keywords: Littlewood class
reviewed by: Ferenc Móricz
MaRDI profile type: Publication
full work available at URL: j.jat.2003.11.004
OpenAlex ID: W2006506801
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DOI: 10.1016/J.JAT.2003.11.004
description: scientific article; zbMATH DE number 2026879