CONDITIONAL INTEGRAL TRANSFORMS AND CONVOLUTIONS OF BOUNDED FUNCTIONS ON AN ANALOGUE OF WIENER SPACE

Abstract

Let <TEX>$C[0,t]$</TEX> denote the function space of all real-valued continuous paths on <TEX>$[0,t]$</TEX>. Define <TEX>$Xn:C[0,t]{\rightarrow}\mathbb{R}^{n+1}$</TEX> and <TEX>$X_{n+1}:C[0,t]{\rightarrow}\mathbb{R}^{n+2}$</TEX> by <TEX>$X_n(x)=(x(t_0),x(t_1),{\cdots},x(t_n))$</TEX> and <TEX>$X_{n+1}(x)=(x(t_0),x(t_1),{\cdots},x(t_n),x(t_{n+1}))$</TEX>, where <TEX>$0=t_0$</TEX> < <TEX>$t_1$</TEX> < <TEX>${\cdots}$</TEX> < <TEX>$t_n$</TEX> < <TEX>$t_{n+1}=t$</TEX>. In the present paper, using simple formulas for the conditional expectations with the conditioning functions <TEX>$X_n$</TEX> and <TEX>$X_{n+1}$</TEX>, we evaluate the <TEX>$L_p(1{\leq}p{\leq}{\infty})$</TEX>-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the functions which have the form <TEX>$${\int}_{L_2[0,t]}{{\exp}\{i(v,x)\}d{\sigma}(v)}{{\int}_{\mathbb{R}^r}}\;{\exp}\{i{\sum_{j=1}^{r}z_j(v_j,x)\}dp(z_1,{\cdots},z_r)$$</TEX> for <TEX>$x{\in}C[0,t]$</TEX>, where <TEX>$\{v_1,{\cdots},v_r\</TEX><TEX>}$</TEX> is an orthonormal subset of <TEX>$L_2[0,t]$</TEX> and <TEX>${\sigma}$</TEX> and <TEX>${\rho}$</TEX> are the complex Borel measures of bounded variations on <TEX>$L_2[0,t]$</TEX> and <TEX>$\mathbb{R}^r$</TEX>, respectively. We then investigate the inverse transforms of the function with their relationships and finally prove that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions, can be expressed in terms of the products of the conditional Fourier-Feynman transforms of each function.