Abstract.It is shown that the integral Hausdorff mean Tp of the FourierStieltjes transform of a measure on the real line is the Fourier transform of an L1 function if and only if Tp. 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 sï%r{x)T p(x) to be the Fourier transform of an L1-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.