Fourier analysis on linear metric spaces

Abstract:

Probability measures on a real complete linear metric space $E$ are studied via their Fourier transform on $E’$ provided $E$ has the approximation property and possesses a real positive definite continuous function $\Phi (x)$ such that $||x|| > \epsilon$ implies $\Phi (0) - \Phi (x) > c(\epsilon )$ where $c(\epsilon ) > 0$. 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 $E’$ to be the Fourier transform of a tight Borel probability on $E$. An extension of the Levy continuity theorem as given by ${\text {L}}$. 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 ${l_p}$ spaces $(0 < p \leqslant 2)$ and hence these results apply to separable Hilbert space.