Conditional Fourier-Feynman transforms and convolutions over continuous paths

Abstract

In the present paper, we evaluate the analytic conditional FourierFeynman transforms and convolution products of bounded functions which are important in Feynman integration theories and quantum mechanics. We then investigate the inverse transforms of the functions with their relationships and finally that the conditional analytic FourierFeynman transforms of the conditional convolution products for the functions, can be expressed in terms of the products of the conditional Fourier-Feynman transform of each function. Mathematics Subject Classification: 28C20

Relationships Between Integral Transforms and Convolutions on an Analogue of Wiener Space

“…Moreover, on C[0, t], the space of real-valued continuous paths on [0, t], Kim [13] extended the relationships between the conditional convolution product and the L p (1 ≤ p ≤ ∞)-analytic conditional Fourier-Feynman transform of the functions in a Banach algebra which corresponds to the CameronStorvick's Banach algebra S [1]. The author and his coauthors [3,4,5,6,7,14] also established relationships between them for various functions on C[0, t]. In particular, he [3] derived an evaluation formula for the L p -analytic conditional Fourier-Feynman transforms and convolution products of unbounded functions with the conditioning functions X n and X n+1 on C[0, t] given by X n (x) = (x(t 0 ), x(t 1 ), · · · , x(t n )) and X n+1 (x) = (x(t 0 ), x(t 1 ), · · · , x(t n ), x(t n+1 )), where n is a positive integer and 0 = t 0 < t 1 < · · · < t n < t n+1 = t is a partition of [0, t], and then, derived their relationships.…”