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On the convergence of \(\sum c_kf(n_kx)\)
István Berkes, Graz University of Technology, Austria, and Michel Weber, Université Louis-Pasteur et C.N.R.S., Strasbourg, France

Memoirs of the American Mathematical Society
2009; 72 pp; softcover
Volume: 201
ISBN-10: 0-8218-4324-9
ISBN-13: 978-0-8218-4324-6
List Price: US$66
Individual Members: US$39.60
Institutional Members: US$52.80
Order Code: MEMO/201/943
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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.

Table of Contents

  • Introduction
  • Mean convergence
  • Almost everywhere convergence: Sufficient conditions
  • Almost everywhere convergence: Necessary conditions
  • Random sequences
  • Discrepancy of random sequences \(\{S_n x\}\)
  • Some open problems
  • Bibliography
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