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 L4 norm of a trigonometric polynomial whose spectrum is a nontrivial strictly monotone polynomial with integer coefficients of degree three and higher, via its L2 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 L4 norm of the trigonometric polynomial in this conjecture.
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BibTeX
@misc{vasilyev2026norm,
title = {{Norm bounds on Fourier series with polynomial spectra and constrained coefficients}},
author = {Ioann Vasilyev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.19132},
eprint = {2608.19132},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19132},
}
RIS
TY - UNPB
TI - Norm bounds on Fourier series with polynomial spectra and constrained coefficients
AU - Ioann Vasilyev
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2608.19132
UR - https://arxiv.org/abs/2608.19132
AB - 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.
ER -