具有多项式谱和约束系数的傅里叶级数的范数界
Norm bounds on Fourier series with polynomial spectra and constrained coefficients
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中文总结 AI 辅助
本文针对谱为多项式的三角多项式，通过系数约束建立了L⁴范数上界，补充了前人成果并解决了特定猜想，确定了对应范数增长的精确阶。
中文摘要 AI 辅助
本文旨在通过L²范数证明一类三角多项式的L⁴范数上界，该多项式的谱为非平凡严格单调的三次及以上整数系数多项式。所研究三角多项式的系数需满足：其构成的复序列模长递减。其次，在谱为完全平方数的情况下，通过对系数施加更严格条件，得到了类似结果。本文结果强化并补充了S. Bochkarev与A. Córdoba的成果，还对Eceizabarrena和Da Rocha的猜想给出了答案，并确定了该猜想中三角多项式L⁴范数增长的精确阶。
英文摘要
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.
发表机构
- St. Petersburg Department of Steklov Mathematical Institute（圣彼得堡斯捷克洛夫数学研究所圣彼得堡分部）
机构由 AI 辅助整理，请以论文原文为准。
