Conditional Fourier-Feynman Transforms and Convolutions of Unbounded Functions on a Generalized Wiener Space

Abstract

Abstract. Let C[0, t] denote the function space of real-valued continuous paths on [0, t]. Define Xn : C[0, t] → R n+1 and X n+1 : C[0, t] → R n+2 by Xn(x) = (x(t 0 ), x(t 1 ), . . . , x(tn)) and X n+1 (x) = (x(t 0 ), x(t 1 ), . . . , x(tn), x(t n+1 )), respectively, where 0 = t 0 < t 1 < · · · < tn < t n+1 = t. In the present paper, using simple formulas for the conditional expectations with the conditioning functions Xn and X n+1 , we evaluate the Lp(1 ≤ p ≤ ∞)-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the functions, which have the form, where {v 1 , . . . , vr} is an orthonormal subset of L 2 [0, t], fr ∈ Lp(R r ), and σ is the complex Borel measure of bounded variation on L 2 [0, t]. We then investigate the inverse conditional Fourier-Feynman transforms of the function and prove that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions can be expressed by the products of the analytic conditional Fourier-Feynman transform of each function.

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

“…Using the same method as used in the proof of Theorems 2.2, 2.3 and 2.4 of [3], we have the following theorems.…”

“…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.…”

“…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. Note that X n is independent of the present positions of the paths in C[0, t] while X n+1 wholly depends on the present positions.…”

Conditional Fourier-Feynman transforms and convolutions over continuous paths

“…Moreover, on C [0, t], the space of real-valued continuous paths on [0, t], Kim [12] 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 Cameron-Storvick's Banach algebra S [1]. The second author [3,4,5,6,7] also established the relationships between them for various functions on C [0, t]. In particular, he [6] derived an evaluation formula for the L p -analytic conditional Fourier-Feynman transforms and convolution products of bounded 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.…”