CONDITIONAL FOURIER-FEYNMAN TRANSFORM AND CONVOLUTION PRODUCT OVER WIENER PATHS IN ABSTRACT WIENER SPACE: AN LpTHEORY

Abstract

In this paper, using a simple formula, we evaluate the conditional Fourier-Feynman transforms and the conditional convolution products of cylinder type functions, and show that the conditional Fourier-Feynman transform of the conditional convolution product is expressed as a product of the conditional Fourier-Feynman transforms. Also, we evaluate the conditional Fourier-Feynman transforms of the functions of the forms exp {<TEX>$\int_{O}^{T}$</TEX> <TEX>$\theta$</TEX>(s,<TEX>$\chi$</TEX>(s))ds}, exp{<TEX>$\int_{O}^{T}$</TEX> <TEX>$\theta$</TEX>(s,<TEX>$\chi$</TEX>(s))ds}<TEX>$\Phi$</TEX>(<TEX>$\chi$</TEX>(T)), exp{<TEX>$\int_{O}^{T}$</TEX> <TEX>$\theta$</TEX>(s,<TEX>$\chi$</TEX>(s))d<TEX>${\zeta}$</TEX>(s)}, exp{<TEX>$\int_{O}^{T}$</TEX> <TEX>$\theta$</TEX>(s,<TEX>$\chi$</TEX>(s))d<TEX>${\zeta}$</TEX>(s)}<TEX>$\Phi$</TEX>(<TEX>$\chi$</TEX>(T)) which are of interest in Feynman integration theories and quantum mechanics.