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Mathematics > Functional Analysis

Title: Integral comparisons of nonnegative positive definite functions on locally compact abelian groups

Abstract: In this paper, we discuss the following general questions. Let $\mu, \nu$ be two regular Borel measures of finite total variation. When do we have a constant $C$ satisfying that $$\int f d\nu \le C \int f d\mu$$ whenever $f$ is a continuous nonnegative positive definite function? How the admissible constants $C$ can be characterized and what is the best value?
First we discuss the problem in locally compact Abelian groups and then apply the results to the case when $\mu, \nu$ are the traces of the usual Lebesgue measure over centered and arbitrary intervals, respectively. This special case was earlier investigated by Shapiro, Montgomery, Hal\'asz and Logan. It is a close companion of the more familiar problem of Wiener, as well.
Comments: 30 pages
Subjects: Functional Analysis (math.FA); Classical Analysis and ODEs (math.CA)
MSC classes: Primary 43A05, 43A35. Secondary 43A25, 43A60, 43A70
Cite as: arXiv:1803.06409 [math.FA]
  (or arXiv:1803.06409v1 [math.FA] for this version)

Submission history

From: Marcell Gábor Gaál [view email]
[v1] Fri, 16 Mar 2018 21:48:01 GMT (38kb)