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The Hausdorff mean of a Fourier-Stieltjes transform

Author: Constantine Georgakis
Journal: Proc. Amer. Math. Soc. 116 (1992), 465-471
MSC: Primary 42A38; Secondary 26D15, 47A35, 47B38
MathSciNet review: 1096210
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Abstract: It is shown that the integral Hausdorff mean $ T\hat \mu $ of the Fourier-Stieltjes transform of a measure on the real line is the Fourier transform of an $ {L^1}$ function if and only if $ T\hat \mu $ vanishes at infinity or the kernel of $ T$ has mean value zero. Also a sufficient condition on the kernel of $ T$ and a necessary and sufficient condition on the measure is established in order for $ - i\operatorname{sign}(x)T\hat \mu (x)$ to be the Fourier transform of an $ {L^1}$-function. These results yield an improvement of Fejer's and Wiener's formulas for the inversion of Fourier-Stieltjes transforms, the uniqueness property of certain generalized Fourier transforms, and a generalization of the mean ergodic theorem for unitary operators.

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Keywords: Hausdorff mean Fourier-Stieltjes, transform, ergodic, unitary operator
Article copyright: © Copyright 1992 American Mathematical Society

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