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# On the Convergence of Sum CKF(NKX)

ISBN-10: 0821843249

ISBN-13: 9780821843246

Pub. Date: 08/07/2009

Publisher: American Mathematical Society

Let $f$ be a periodic measurable function and $(n_k)$ an increasing sequence of positive integers. The authors study conditions under which the series $\sum_{k=1}^\infty c_k f(n_kx)$ converges in mean and for almost every $x$. There is a wide classical literature on this problem going back to the 30's, but the results for general $f$ are much less complete than in

## Overview

Let $f$ be a periodic measurable function and $(n_k)$ an increasing sequence of positive integers. The authors study conditions under which the series $\sum_{k=1}^\infty c_k f(n_kx)$ converges in mean and for almost every $x$. There is a wide classical literature on this problem going back to the 30's, but the results for general $f$ are much less complete than in the trigonometric case $f(x)=\sin x$. As it turns out, the convergence properties of $\sum_{k=1}^\infty c_k f(n_kx)$ in the general case are determined by a delicate interplay between the coefficient sequence $(c_k)$, the analytic properties of $f$ and the growth speed and number-theoretic properties of $(n_k)$. In this paper the authors give a general study of this convergence problem, prove several new results and improve a number of old results in the field. They also study the case when the $n_k$ are random and investigate the discrepancy the sequence $\{n_kx\}$ mod 1.

## Product Details

ISBN-13:
9780821843246
Publisher:
American Mathematical Society
Publication date:
08/07/2009
Series:
Memoirs of the American Mathematical Society Series , #201
Pages:
72
Product dimensions:
6.60(w) x 9.80(h) x 0.20(d)

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