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

Auteurs : Philippe Jaming
, Chadi Saba,  

Revue : Advances in Pure and Applied Mathematics 

Numéro : Issue 3 (June 2024) 

Volume : 15 

Date : 2024/06/12 

DOI : 10.21494/ISTE.OP.2024.1173 

ISSN : 1869-6090 

Résumé : 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. 

Éditeur : ISTE OpenScience