The Wiener closure theorems for abstract Wiener spaces

Kuelbs, J.; Mandrekar, V.

doi:10.1090/S0002-9939-1972-0293053-8

The Wiener closure theorems for abstract Wiener spaces
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by J. Kuelbs and V. Mandrekar PDF

Proc. Amer. Math. Soc. 32 (1972), 169-178 Request permission

Abstract:

We introduce ${\mathcal {L}_1}$ and ${\mathcal {L}_2}$ translates for functions in ${\mathcal {L}_1}(\mu )$ and ${\mathcal {L}_2}(\mu )$ where $\mu$ is a Gaussian measure on a Banach space. With these translates and the Fourier-Wiener transforms defined by Cameron and Martin we obtain Wiener’s closure theorem in ${\mathcal {L}_2}(\mu )$ and in ${\mathcal {L}_1}(\mu )$. Using the ${\mathcal {L}_1}(\mu )$ results we indicate the analogue of the Wiener-Pitt Tauberian theorems for this setup.

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