Bulletin of the
Korean Mathematical Society BKMS

Using a simple formula for conditional expectations over an analogue of Wiener space, we calculate a generalized analytic conditional Fourier-Feynman transform and convolution product of generalized cylinder functions which play important roles in Feynman integration theories and quantum mechanics. We then investigate their relationships, that is, the conditional Fourier-Feynman transform of the convolution product can be expressed in terms of the product of the conditional Fourier-Feynman transforms of each function. Finally we establish change of scale formulas for the generalized analytic conditional Fourier-Feynman transform and the conditional convolution product. In this evaluation formulas and change of scale formulas we use multivariate normal distributions so that the orthonormalization process of projection vectors which are essential to establish the conditional expectations, can be removed in the existing conditional Fourier-Feynman transforms, conditional convolution products and change of scale formulas.

Keywords: analogue of Wiener space, analytic conditional Fourier-Feynman transform, change of scale formula, conditional convolution product, Wiener space

MSC numbers: Primary 28C20; Secondary 60H05, 60G15