Spectral factorization of trigonometric polynomials and lattice geometry
type: Item
instance of: scholarly article
title: Spectral factorization of trigonometric polynomials and lattice geometry
zbMATH Open document ID: 1286.42001
published in: Acta Arithmetica
publication date: 2012-10-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1208.5590
Mathematics Subject Classification ID: 42A05
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 42B05
zbMATH DE Number: 6090473
zbMATH Keywords: Fejér-Riesz spectral factorization
zbMATH Keywords: Hilbert kernel
zbMATH Keywords: Wiener algebra
zbMATH Keywords: Arens-Roydon theorem
zbMATH Keywords: modular group
author: Wayne Lawton
MaRDI profile type: Publication
OpenAlex ID: W2963059593
arXiv ID: 1208.5590
DOI: 10.4064/AA155-3-9
review text: It was shown by \textit{F. Riesz} [J. Reine Angew. Math. 146, 83--87 (1915; JFM 45.0408.01)] that if \(f(X)\) is a non-negative trigonometric polynomial having frequencies in the difference set \(F-F\), where \(F\) is an interval \([0,n]\), then \(f(X)=|g(X)|^2\), where \(g(X)\) is a trigonometrical polynomial with frequencies in \(F\). The author conjectures the following analogue of this result: if \(\alpha,\beta\) are real, and \(F\) is the set of lattice points \((x,y)\) in the region bounded by the lines \(\alpha x+\beta,\alpha x-\beta\), and \(f(X,Y)\) is a non-negative trigonometric polynomial having frequencies in \(F-F\), then \(f\) is a uniform limit of a sequence \(|f_n(X,Y)|^2\), where \(f_n\) are trigonometric polynomials with frequencies in \(F\). He proves this conjecture in the cases when \(\alpha\) is either rational or there exists a sequence of matrices \(g^{(n)}\in SL_2(Z)\) such that NEWLINE\[NEWLINE\lim_{n\to\infty}|g{(n)}_{21}+\alpha g{(n)}_{22}||g{(n)}_{22}|\log(|g{(n)}_{22}|)=0.NEWLINE\]
reviewed by: Władysław Narkiewicz
description: scientific article; zbMATH DE number 6090473