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Generalizations of the Sidon-Telyakovskii theorem. (English) Zbl 0647.42007

S. A. Telyakovskij [(*) Math. Notes 14, 742-748 (1974; Zbl 0283.42011)] proved the theorem: The class S of real null sequences is defined as follows: A real null sequence \(\{a_ n\}\) belongs to S if there exists a monotone sequence \(\{A_ m\}\) such that \(\sum^{\infty}_{n=1}A_ n<\infty\) and (1) \(| \Delta a_ n| \leq A_ n\) for all n. In this paper the authors replace the condition (1) of Telyakovskij (*) by the condition \((2)\quad (1/n)\sum^{n}_{k=1}| \Delta c(k)|^ p/n^ p_ k=O(1),\) \(n\to \infty\), which is weaker than condition (1), where \(p>1\), \(\{A_ n\}\) is a monotone sequence such that \(\sum^{\infty}_{n=1}A_ n<\infty\) and \(\{\) c(n)\(\}\) is a null sequence of complex numbers. The authors have left open many problems for future workers and interesting results can be established on this line.
Reviewer: S.Sharma

MSC:

42A20 Convergence and absolute convergence of Fourier and trigonometric series
42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)

Citations:

Zbl 0283.42011
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