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

2023/11/29 by Philippe Jaming, Jaming, Philippe, Chadi Saba +1 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.2311.17714

openalex publication_date 2023/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• On a Conjecture of Littlewood and Idempotent Measures

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• Extremizers of a Fourier uncertainty principle related to averaging

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