# On the convergence of $\sum c_kf(n_kx)$

### About this Title

**István Berkes** and **Michel Weber**

Publication: Memoirs of the American Mathematical Society

Publication Year
2009: Volume 201, Number 943

ISBNs: 978-0-8218-4324-6 (print); 978-1-4704-0557-1 (online)

DOI: http://dx.doi.org/10.1090/memo/0943

MathSciNet review: 2541755

MSC (2000): Primary 42C15; Secondary 11K38, 30B50, 42A55, 60G50

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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

**Chapters**

- Introduction
- Chapter 1. Mean convergence
- Chapter 2. Almost everywhere convergence: Sufficient conditions
- Chapter 3. Almost everywhere convergence: Necessary conditions
- Chapter 4. Random sequences
- Chapter 5. Discrepancy of random sequences $\{S_nx\}$
- Chapter 6. Some open problems