Proceedings of the Steklov Institute of Mathematics ( IF 0.4 ) Pub Date : 2021-04-22 , DOI: 10.1134/s0081543821010193 Sergey Yu. Tikhonov
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:
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。