The Hausdorff Mean of a Fourier-Stieltjes Transform
Constantine Georgakis · Proceedings of the American Mathematical Society · 1992
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.