Abstract
In this paper, using the concept of a generalized Feynman integral, we define a generalized Fourier‐Feynman transform and a generalized convolution product. Then for two classes of functionals on Wiener space we obtain several results involving and relating these generalized transforms and convolutions. In particular we show that the generalized transform of the convolution product is a product of transforms. In addition we establish a Parseval′s identity for functionals in each of these classes.
Highlights
The concept of an L 1 analytic Fourier-Feynman transform (FFT) was introduced by Brue in [1]
Both the FFT and the convolution product are defined in terms of a Feynman integral
In our theorem we show that the GFFT of the generalized convolution product (GCP) is the product of transforms
Summary
The concept of an L 1 analytic Fourier-Feynman transform (FFT) was introduced by Brue in [1]. In [4], Huffman, Park and Skoug defined a convolution product for functionals on Wiener space and in [4,5] obtained various results involving the FFT and the convolution product. Both the FFT and the convolution product are defined in terms of a Feynman integral. In defining the FFT [1,2,3] of F and the convolution product [4] of F and G, one starts with, for A > 0, the Wiener integrals ol. We define the (generalized) analytic Feynman integral of F with parameter q by anfq anA.
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