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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Fourier analysis on linear metric spaces

Author: J. Kuelbs
Journal: Trans. Amer. Math. Soc. 181 (1973), 293-311
MSC: Primary 60B10; Secondary 28A40, 43A35
MathSciNet review: 0331455
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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 $ \vert\vert x\vert\vert > \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.

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PII: S 0002-9947(1973)0331455-7
Keywords: Fourier transform, Bochner's theorem, Levy continuity theorem, cylinder set measure, metric approximation property, basis, Orlicz spaces, tight measures
Article copyright: © Copyright 1973 American Mathematical Society