exit

Mathématiques   > Accueil   > Avancées en Mathématiques Pures et Appliquées   > Numéro 3 (Juin 2024)   > Article

D’Ingham à l’inégalité de Nazarov : un survey sur quelques inégalités trigonométriques

From Ingham to Nazarov’s inequality: a survey on some trigonometric inequalities


Philippe Jaming
Univ. Bordeaux
France

Chadi Saba
Univ. Bordeaux
France



Publié le 12 juin 2024   DOI : 10.21494/ISTE.OP.2024.1173

Résumé

Abstract

Mots-clés

Keywords

The aim of this paper is to give an overview of some inequalities about $$$L^p$$$-norms ($$$p$$$ = 1 or $$$p$$$ = 2) of harmonic (periodic) and non-harmonic trigonometric polynomials. Among the material covered, we mention Ingham’s Inequality about $$$L^2$$$ norms of non-harmonic trigonometric polynmials, the proof of the Littlewood conjecture by McGehee, Pigno and Smith on the lower bound of the $$$L^1$$$ norm of harmonic trigonometric polynomials as well as its counterpart in the non-harmonic case due to Nazarov. For the later one, we give a quantitative estimate that completes our recent result with an estimate of $$$L^1$$$-norms over small intervals. We also give some stronger lower bounds when the frequencies satisfy some more restrictive conditions (lacunary Fourier series, “multi-step arithmetic sequences”). Most proofs are close to existing ones and some open questions are mentionned at the end.

The aim of this paper is to give an overview of some inequalities about $$$L^p$$$-norms ($$$p$$$ = 1 or $$$p$$$ = 2) of harmonic (periodic) and non-harmonic trigonometric polynomials. Among the material covered, we mention Ingham’s Inequality about $$$L^2$$$ norms of non-harmonic trigonometric polynmials, the proof of the Littlewood conjecture by McGehee, Pigno and Smith on the lower bound of the $$$L^1$$$ norm of harmonic trigonometric polynomials as well as its counterpart in the non-harmonic case due to Nazarov. For the later one, we give a quantitative estimate that completes our recent result with an estimate of $$$L^1$$$-norms over small intervals. We also give some stronger lower bounds when the frequencies satisfy some more restrictive conditions (lacunary Fourier series, “multi-step arithmetic sequences”). Most proofs are close to existing ones and some open questions are mentionned at the end.

Ingham’s Inequality Littlewood problem non-harmonic Fourier series lacunary series

Ingham’s Inequality Littlewood problem non-harmonic Fourier series lacunary series