The Hausdorff mean of a Fourier-Stieltjes transform

Georgakis, Constantine

doi:10.1090/S0002-9939-1992-1096210-9

The Hausdorff mean of a Fourier-Stieltjes transform
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Proc. Amer. Math. Soc. 116 (1992), 465-471 Request permission

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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