eBook ISBN:  9781470405571 
Product Code:  MEMO/201/943.E 
List Price:  $66.00 
MAA Member Price:  $59.40 
AMS Member Price:  $39.60 
eBook ISBN:  9781470405571 
Product Code:  MEMO/201/943.E 
List Price:  $66.00 
MAA Member Price:  $59.40 
AMS Member Price:  $39.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 201; 2009; 72 ppMSC: Primary 42; 30; 11; 60
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 numbertheoretic 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


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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 numbertheoretic 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.

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