Abstract

We refine the classical boundedness criterion for sums of sine series with monotone coefficients \(b_k\): the sum of a series is bounded on \(\mathbb R\) if and only if the sequence \({\{kb_k\}}\) is bounded. We derive a two-sided estimate of the Chebyshev norm of the sum of a series via a special norm of the sequence \(\{kb_k\}\). The resulting upper bound is sharp, and the constant in the lower bound differs from the exact value by at most \(0.2\).

中文翻译：

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带有单调系数的正弦系列之和的$$ L ^ \ infty $$范数的两面估计$$ \ {^ _ infty $$-序列$$ \ {kb_k \} $$

摘要

我们针对具有单调系数\（b_k \）的正弦序列之和完善经典有界性准则：当且仅当序列\（{\ {kb_k \}}时，序列之和以\（\ mathbb R \）为界\）是有界的。通过序列\（\ {kb_k \} \）的特殊范数，我们得出了一系列总和的切比雪夫范数的双面估计。所得的上限是尖锐的，并且下限中的常数与精确值的差异最多为\（0.2 \）。