On the Convolution Powers of Complex Functions on \(\mathbb {Z}\)

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• 作者：Evan Randles ; Laurent Saloff-Coste
• 关键词：Convolution powers ; Local limit theorems ; Approximation ; Primary 42A85 ; Secondary 60F99
• 刊名：Journal of Fourier Analysis and Applications
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：21
• 期：4
• 页码：754-798
• 全文大小：1,820 KB
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• 作者单位：Evan Randles (1)
  Laurent Saloff-Coste (2)
  
  1. Center for Applied Mathematics, Cornell University, Ithaca, USA
  2. Department of Mathematics, Cornell University, Ithaca, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Fourier Analysis
  Abstract Harmonic Analysis
  Approximations and Expansions
  Partial Differential Equations
  Applications of Mathematics
  Signal,Image and Speech Processing
  
• 出版者：Birkh盲user Boston
• ISSN：1531-5851

文摘

The local limit theorem describes the behavior of the convolution powers of a probability distribution supported on \(\mathbb {Z}\). In this work, we explore the role played by positivity in this classical result and study the convolution powers of the general class of complex valued functions finitely supported on \(\mathbb {Z}\). This is discussed as de Forest’s problem in the literature and was studied by Schoenberg and Greville. Extending these earlier works and using techniques of Fourier analysis, we establish asymptotic bounds for the sup-norm of the convolution powers and prove extended local limit theorems pertaining to this entire class. As the heat kernel is the attractor of probability distributions on \(\mathbb {Z}\), we show that the convolution powers of the general class are similarly attracted to a certain class of analytic functions which includes the Airy function and the heat kernel evaluated at purely imaginary time.