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On the convergence of $\sum c_kf(n_kx)$
 
István Berkes Graz University of Technology, Graz, Austria
Michel Weber Université Louis-Pasteur et C.N.R.S., Strasbourg, France
On the convergence of sumc_kf(n_kx)
eBook ISBN:  978-1-4704-0557-1
Product Code:  MEMO/201/943.E
List Price: $66.00
MAA Member Price: $59.40
AMS Member Price: $39.60
On the convergence of sumc_kf(n_kx)
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On the convergence of $\sum c_kf(n_kx)$
István Berkes Graz University of Technology, Graz, Austria
Michel Weber Université Louis-Pasteur et C.N.R.S., Strasbourg, France
eBook ISBN:  978-1-4704-0557-1
Product Code:  MEMO/201/943.E
List Price: $66.00
MAA Member Price: $59.40
AMS Member Price: $39.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2012009; 72 pp
    MSC: 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 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2012009; 72 pp
MSC: 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 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.

  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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