Analytic Fourier-Feynman transforms and convolution

Abstract

Abstract. In this paper we develop an Lp Fourier-Feynman theory for a class of functionals on Wiener space of the form F(x) = f(J0 axdx, ... , /0 a"dx). We then define a convolution product for functionals on Wiener space and show that the Fourier-Feynman transform of the convolution product is a product of Fourier-Feynman transforms.

Relationship Between the Wiener Integral and the Analytic Feynman Integral of Cylinder Function

“…We close this section by introducing the analytic Wiener and the analytic Feynman integrals of the cylinder function. One can find the results in, for example, [6].…”

Parts Formulas Involving Conditional Integral Transforms on Function Space

“…In [10], the current authors and Skoug showed that for all F and G in E σ , F α,β (F X) and ((F * G) α X) exist and belong to E σ for all nonzero complex numbers α and β and the condition by X(x) = x(T ) while δF (y|w) exists and belongs to E σ for all y and w in K. For related work see [2,7,9,11,15] and for a detailed survey of previous work see [14].…”

“…Definition 1.2. Let F and G be functionals defined on K. Then the convolution product (F * G) α of F and G is defined by [7,9,13,15]. Definition 1.3.…”

Generalized convolution product for an integral transform on a Wiener space

“…For certain values of the parameters γ and η and for certain classes of functionals, the Fourier-Wiener transform [1], the Fourier-Feynman transform [2,8], and the Gauss transform [10] are special cases of his integral transform F γ,η .…”