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arXiv · 2608.19132

Norm bounds on Fourier series with polynomial spectra and constrained coefficients

Ioann Vasilyev
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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. C\'ordoba. 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.

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Ioann Vasilyev. 2026-08-19. Norm bounds on Fourier series with polynomial spectra and constrained coefficients. https://arxiv.org/abs/2608.19132

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