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

Let C[0; t] denote the function space of all real-valued continuous paths on [0; t]. De&macr;ne Xn : C[0; t] ! Rn+1 and Xn+1 : C[0; t] ! Rn+2 by Xn(x) = (x(t0); x(t1); &cent; &cent; &cent; ; x(tn)) and Xn+1(x) =(x(t0); x(t1); &cent; &cent; &cent; ; x(tn); x(tn+1)), where 0 = t0 < t1 < &cent; &cent; &cent; < tn <tn+1 = t. In the present paper, using simple formulas for the conditional expectations with the conditioning functions Xn and Xn+1, we evaluate the Lp(1 · p · 1)-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the functions which have the form Z L2[0;t] expfi(v; x)gd¾(v) ZRr exp ½i Xrj=1zj(vj ; x)¾d½(z1; &cent; &cent; &cent; ; zr) for x 2 C[0; t], where fv1; &cent; &cent; &cent; ; vrg is an orthonormal subset of L2[0; t] and ¾ and ½ are the complex Borel measures of bounded variations on L2[0; t] and Rr, respectively. We then investigate the inverse transforms of the function with their relationships and&macr;nally prove that the analytic conditional Fourier-Feynman trans-forms 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.