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Fourier Transforms of Unbounded Measures

Published online by Cambridge University Press:  20 November 2018

James Stewart
James Stewart*
Affiliation:
McMaster University, Hamilton, Ontario
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1. Introduction. One of the basic objects of study in harmonic analysis is the Fourier transform (or Fourier-Stieltjes transform) μ of a bounded (complex) measure μ on the real line R:

(1.1)

More generally, if μ is a bounded measure on a locally compact abelian group G, then its Fourier transform is the function

(1.2)

where Ĝ is the dual group of G and One answer to the question “Which functions can be represented as Fourier transforms of bounded measures?” was given by the following criterion due to Schoenberg [11] for the real line and Eberlein [5] in general: f is a Fourier transform of a bounded measure if and only if there is a constant M such that

(1.3)

for all ϕ ∈ L1(G) where

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 6 , 01 December 1979 , pp. 1281 - 1292
Copyright
Copyright © Canadian Mathematical Society 1979

References

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