Abstract

We study the trigonometric series \(\sum_{n=1}^\infty \lambda_n \cos nx\) and \(\sum_{n=1}^\infty \lambda_n \sin nx\) with \(\{\lambda_n\}\) being a sequence of bounded variation. Let \(\psi\) denote the sum of such a series. We obtain necessary and sufficient conditions for the validity of the weighted Fourier inequality \(\left(\intop_0^\pi\mathopen|\psi(x)|^q \omega(x)\,dx\right)^{1/q} \le C\!\left(\sum_{n=1}^\infty u_n\left(\sum_{k=n}^\infty\mathopen|\lambda_{k}-\lambda_{k+1}|\right)^p \right)^{1/p}\), \(0<p\le q<\infty\), in terms of the weight \(\omega\) and the weighted sequence \(\{u_n\}\). Applications to the series with general monotone coefficients are given.

中文翻译：

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加权傅立叶不等式和变异的有界性

摘要

我们研究了三角级数\（\ sum_ {N = 1} ^ \ infty \ lambda_n \ COS NX \）和\（\ sum_ {N = 1} ^ \ infty \ lambda_n \罪NX \）与\（\ {\ lambda_n \} \）是有界变异的序列。令\（\ psi \）表示这样一个系列的总和。我们获得了加权傅立叶不等式\（\ left（\ intop_0 ^ \ pi \ mathopen | \ psi（x）| ^ q \ omega（x）\，dx \ right）^ {1 /的有效性的充分必要条件q} \ le C \！\ left（\ sum_ {n = 1} ^ \ infty u_n \ left（\ sum_ {k = n} ^ \ infty \ mathopen | \ lambda_ {k}-\ lambda_ {k + 1} | \ right）^ p \ right）^ {1 / p} \），\（0 <p \ le q <\ infty \），根据权重\（\ omega \）和加权序列\（\ {联合国\}\）。给出了具有一般单调系数的级数的应用。