Illinois Journal of Mathematics

On continuity of the variation and the Fourier transform

P. H. Maserick

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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.

Article information

Illinois J. Math., Volume 29, Issue 2 (1985), 302-310.

First available in Project Euclid: 20 October 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 43A35: Positive definite functions on groups, semigroups, etc.
Secondary: 44A60: Moment problems


Maserick, P. H. On continuity of the variation and the Fourier transform. Illinois J. Math. 29 (1985), no. 2, 302--310. doi:10.1215/ijm/1256045731.

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