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Integral transforms of functionals in \(L_2(C_0[0,T])\). (English) Zbl 1062.28017

In [J. Funct. Anal. 47, 153–164 (1982; Zbl 0487.44006)], Y. Lee introduced an integral transform of functionals on an abstract Wiener space which is an extension of the Fourier-Wiener transform on an abstract Wiener space. He applied this transform to differential equations on infinite dimensional spaces. In [Numer. Funct. Anal. Optimization 21, No. 1–2, 97–105 (2000; Zbl 0948.28008)], the first author, I. Yoo and the reviewer established the relationship between the integral transforms of exponential type of analytic functionals on an abstract Wiener space and the integral transform of their convolution.
In this paper, the authors give a necessary and sufficient condition that a square integrable functional on a Wiener space has an integral transform which is also square integrable on a Wiener space.

MSC:

28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
60J65 Brownian motion
44A15 Special integral transforms (Legendre, Hilbert, etc.)
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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