A Time-Independent Conditional Fourier-Feynman Transform and Convolution Product on an Analogue of Wiener Space

Abstract

Abstract. Let C[0, t] denote the function space of all real-valued continuous paths on [0, t]. Define Xn :, where 0 = t0 < t1 < · · · < tn < t is a partition of [0, t]. In the present paper, using a simple formula for the conditional expectation given the conditioning function Xn, we evaluate the Lp(1 ≤ p ≤ ∞)-analytic conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions which have the formwhere {v1, · · · , vr} is an orthonormal subset of L2[0, t] and f ∈ Lp(R r ). We then investigate several relationships between the conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions.

Conditional Fourier-Feynman transforms and convolutions over continuous paths

“…where H 3 is given by (5). By (4), the Morera's theorem and the dominated convergence theorem, we have the existence of…”

“…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.…”

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

“…Using the same method as used in the proof of Theorem 3.2 in [5] with Theorems 2.4, 3.5, (2.4) and (3.5), we can prove the following theorems. Theorem 4.3.…”

“…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.…”

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

“…The author [9] also did the same on the relationships between the convolution and the transform for the products of the functions in S wϕ and the bounded cylinder functions of the Fourier-Stieltjes transforms of measures on the Borel class of R r . Furthermore, he [7,8,10] established several relationships between the L p -analytic conditional FourierFeynman transforms and the conditional convolution products of the cylinder functions on C[0, t]. In particular, he [7,8] derived evaluation formulas for the L p -analytic conditional Fourier-Feynman transforms and the conditional convolution products of the same cylinder functions with the conditioning functions X n : C[0, t] → R n+1 and X n+1 : C[0, t] → R n+2 given by X n (x) = (x(t 0 ), x(t 1 ), .…”