Abstract
In this paper, we recall a more generalized integral transform, a generalized convolution product and a generalized first variation on function space. The Gaussian process and the bounded linear operators on function space are used to define them. We then establish the existence and various relationships between the generalized integral transform and the generalized convolution product. Furthermore, we obtain some relationships between the generalized integral transform and the generalized first variation with the generalized Cameron–Storvick theorem. Finally, some applications are demonstrated as examples.
Highlights
For T > 0, let C0 [0, T ] be the one-parameter Wiener space and let M denote the class of all Wiener measurable subsets of C0 [0, T ]
In [1], Lee studied an integral transform of analytic functionals on abstract Wiener spaces
We start this section by giving definition of generalized integral transform, generalized convolution product and the generalized first variation of functionals on K
Summary
For T > 0, let C0 [0, T ] be the one-parameter Wiener space and let M denote the class of all Wiener measurable subsets of C0 [0, T ]. In [1], Lee studied an integral transform of analytic functionals on abstract Wiener spaces. Researchers have studied some theories of integral transform for functionals on function space. If h, h1 and h2 are identically 1 on [0, T ], Equations (2) and (3) reduce to Equation (1) Another method is using the operators on K. In [6,16], the authors used this operators to generalize the integral transforms. Many relationships among the integral transform, the convolution and the first variation have been obtained. We use the both concepts, the Gaussian process and the operator, to define a more generalized integral transform, a generalized convolution product and a generalized first variation of functionals on function space. By choosing the kernel functions and operators, all results and formulas in previous papers are corollaries of our results and formulas in this paper
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