Norm bounds on Fourier series with polynomial spectra and constrained coefficients

Abstract

The goal of this paper is to prove an upper bound for the $L^4$ norm of a trigonometric polynomial whose spectrum is a nontrivial strictly monotone polynomial with integer coefficients of degree three and higher, via its $L^2$ norm. Our condition on the coefficients of the trigonometric polynomial in question is that they form a complex sequence whose modulus is decreasing. Second, we obtain a similar result in the case where the spectrum is formed by perfect squares, under a more strict condition on the coefficients. Our results strengthen and complement those by S. Bochkarev and A. Córdoba. We also give an answer to a conjecture of Eceizabarrena and Da Rocha and determine the sharp order of growth of the $L^4$ norm of the trigonometric polynomial in this conjecture.

Disclosure

“N 2 j=0 and we are done. Acknowledgments The author is grateful to Sergey Kislyakov and to Mikhail Vasilyev for a number of helpful discussions. The author acknowledges the use of ChatGPT 5.6 Plus (OpenAI) as a research tool in developing and checking some technical parts of the proof of Theorem 3. Namely: in the arithmetic estimates used in the argument, and in locating relevant references. The author has independently che”

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Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-22T18:19:02.476363Z

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