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Summer 1985 On continuity of the variation and the Fourier transform
P. H. Maserick
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Illinois J. Math. 29(2): 302-310 (Summer 1985). DOI: 10.1215/ijm/1256045731


Let $S$ be a commutative semitopological semigroup with identity and involution, $\Gamma$ a compact subset in the topology of pointwise convergence of the set of semicharacters on $S$. Let $f$ be a function which admits a (necessarily unique) integral representation of the form $$f(s)=\int_{\Gamma}{\rho(s)d\mu_{f}(\rho)}\quad (\rho \in \Gamma,s \in S$$ with respect to a complex regular Borel measure $\mu_{f}$ on $\Gamma$. The function $|f|(\cdot)$ defined by $|f|(s)=\int_{\Gamma}{\rho(s)d|\mu_{f}|}$ is called the variation of $f$. It is shown that the variation $|f|$ is bounded and continuous if and only if $f$ is also bounded and continuous. This, coupled with the author's previous characterization of functions of bounded variation, gives a new description of the Fourier transforms of bounded measures on locally compact commutative groups.


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P. H. Maserick. "On continuity of the variation and the Fourier transform." Illinois J. Math. 29 (2) 302 - 310, Summer 1985.


Published: Summer 1985
First available in Project Euclid: 20 October 2009

zbMATH: 0546.43005
MathSciNet: MR784525
Digital Object Identifier: 10.1215/ijm/1256045731

Primary: 43A35
Secondary: 44A60

Rights: Copyright © 1985 University of Illinois at Urbana-Champaign

Vol.29 • No. 2 • Summer 1985
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