Integral transforms, convolution products, and first variations

نویسندگان

Bongjin Kim

Byoung Soo Kim

David L. Skoug

چکیده

We establish the various relationships that exist among the integral transform Ᏺ α,β F , the convolution product (F * G) α , and the first variation δF for a class of functionals defined on K[0,T ], the space of complex-valued continuous functions on [0,T ] which vanish at zero. 1. Introduction and definitions. In a unifying paper [10], Lee defined an integral transform Ᏺ α,β of analytic functionals on an abstract Wiener space. For certain values of the parameters α and β and for certain classes of functionals, the Fourier-Wiener transform [2], the Fourier-Feynman transform [3], and the Gauss transform are special cases of his integral transform Ᏺ α,β. In [5], Chang et al. established an interesting relationship between the integral transform and the convolution product for functionals on an abstract Wiener space. In this paper, we study the relationships that exist among the integral transform, the convolution product, and the first variation [1, 4, 9, 11]. Let C 0 [0,T ] denote one-parameter Wiener space, that is, the space of all real-valued continuous functions x(t) on [0,T ] with x(0) = 0. Let ᏹ denote the class of all Wiener measurable subsets of C 0 [0,T ] and let m denote Wiener measure. Then (C 0 [0,T ],ᏹ, m) is a complete measure space and we denote the Wiener integral of a Wiener inte-grable functional F by

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