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On Fourier-Stieltjes Transforms

Published online by Cambridge University Press:  20 November 2018

A. P. Calderón  and
A. Devinatz
A. P. Calderón
Affiliation:
Institute for Advanced Study
A. Devinatz
Affiliation:
Institute for Advanced Study
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Let be the class of bounded non-decreasing functions defined on the real line which are normalized by the conditions ϕ(− ∞) = 0 , ϕ(t + 0) = ϕ(t).

Let be the class of Fourier-Stieltjes transforms of elements of i.e. the elements of and are connected by the relation

where ϕ ∊ and Φ ∊ .It is well known, and easy to verify that this mapping from to is one to one (1, p. 67, Satz 18).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 7 , 1955 , pp. 453 - 461
Copyright
Copyright © Canadian Mathematical Society 1955

References

1. Bochner, S., Vorlesungen über Fouriersche Integrale (Leipzig, 1932).Google Scholar
2. Caratheodory, C., Vorlesungen über reelle Funktionen (Leipzig und Berlin, 1918).Google Scholar
3. Dyson, F. J., Fourier transforms of distribution functions, Can. J. Math., 5 (1953), 554–558.Google Scholar
4. Saks, S., Theory of the integral (New York, 1937).Google Scholar