# On the convergence of \(∑c_{k}f(n_{k}x)\)

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*István Berkes; Michel Weber*

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

# Table of Contents

## On the convergence of $c_{k}f(n_{k}x)$

- Contents v6 free
- Introduction 110 free
- Chapter 1. Mean convergence 716 free
- Chapter 2. Almost everywhere convergence: sufficient conditions 1726
- Chapter 3. Almost everywhere convergence: necessary conditions 3948
- Chapter 4. Random sequences 4958
- Chapter 5. Discrepancy of random sequences {S[sub(n)]x} 6372
- Chapter 6. Some open problems 6978
- Bibliography 7180