Fourier analysis on linear metric spaces

Abstract

Probability measures on a real complete linear metric space E E are studied via their Fourier transform on E ′ E’ provided E E has the approximation property and possesses a real positive definite continuous function Φ ( x ) \Phi (x) such that | | x | | > ϵ ||x|| > \epsilon implies Φ ( 0 ) − Φ ( x ) > c ( ϵ ) \Phi (0) - \Phi (x) > c(\epsilon ) where c ( ϵ ) > 0 c(\epsilon ) > 0 . In this setting we obtain conditions on the Fourier transforms of a family of tight Borel probabilities which yield tightness of the family of measures. This then is applied to obtain necessary and sufficient conditions for a complex valued function on E ′ E’ to be the Fourier transform of a tight Borel probability on E E . An extension of the Levy continuity theorem as given by L {\text {L}} . Gross for a separable Hilbert space is obtained for such metric spaces. We also prove that various Orlicz-type spaces are in the class of spaces to which our results apply. Finally we apply our results to certain Orlicz-type sequence spaces and obtain conditions sufficient for tightness of a family of probability measures in terms of uniform convergence of the Fourier transforms on large subsets of the dual. We also obtain a more explicit form of Bochner’s theorem for these sequence spaces. The class of sequence spaces studied contains the l p {l_p} spaces ( 0 > p ⩽ 2 ) (0 > p \leqslant 2) and hence these results apply to separable Hilbert space.