Integral Transforms of Functionals in L 2(C a,b [0,T])

Abstract

In this paper we use generalized Fourier-Hermite functionals to obtain a complete orthonormal set inis a very general function space. We then proceed to give a necessary and sufficient condition that a functional F in L 2 (C a,b [0, T ]) has an integral transform F γ,β F also belonging to L 2 (C a,b [0, T ]).

Some basic relationships among transforms, convolution products, first variations and inverse transforms

“…denote the Paley-Wiener-Zygmund stochastic integral [5,7,8,10]. Then is a Gaussian random variable with mean ).…”

“…For example fix σ ∈ (0 1) and let F and G be elements of E σ . Then (F * G) γ is given by (8) and so, by (11) and the inequality…”

“…In [8], Chang, Chung and Skoug gave a necessary and sufficient condition that a functional F in L 2 (C [0 T ]) has an integral transform F γ β F also belonging to L 2 (C [0 T ]).…”

Some Expressions for the Inverse Integral Transform via the Translation Theorem on Function Space

“…The function space C a,b [0, T ] induced by a generalized Brownian motion was introduced by J. Yeh in [20] and was used extensively in [5,7,8,9,10]. In [5], the authors gave a necessary and sufficient condition that a functional F in L 2 (C a,b [0, T ]) has an integral transform F γ,β F also belonging to L 2 (C a,b [0, T ]).…”

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Abstract.In this paper, we analyze the necessary and sufficient condition introduced in [5]. We then establish the inverse integral transforms of the functionals in L 2 (C a,b [0, T ]) and then examine various properties with respect to the inverse integral transforms via the translation theorem.

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