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Pré-Publication, Document De Travail Année : 2023

FROM INGHAM TO NAZAROV'S INEQUALITY: A SURVEY ON SOME TRIGONOMETRIC INEQUALITIES

Chadi Saba

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 2 norms of non-harmonic trigonometric polynomials, the proof of the Littlewood conjecture by Mc Gehee, Pigno and Smith on the lower bound of the 1 norm of harmonic trigonometric polynomials as well as its counterpart in the non-harmonic case due to Nazarov. For the latter one, we give a quantitative estimate that completes our recent result with an estimate of 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 mentioned at the end.
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Dates et versions

hal-04314308 , version 1 (29-11-2023)

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Philippe Jaming, Chadi Saba. FROM INGHAM TO NAZAROV'S INEQUALITY: A SURVEY ON SOME TRIGONOMETRIC INEQUALITIES. 2023. ⟨hal-04314308⟩

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