Relationships among the First Variation, the Convolution Product, and the Fourier-Feynman Transform

Integral transforms, convolution products, and first variations

“…In Section 3, we examine all relationships involving exactly two of the three concepts of "integral transform," "convolution product," and "first variation," while in Section 4, we examine all relationships involving all three of these concepts where each concept is used exactly once. For related work, see [2,5,7,9,10,11,13,14] and for a detailed survey of previous work, see [12]. Remark 1.4.…”

“…Note that if E 0 is given by (1.7), then the entire function f (λ 1 ,...,λ n ) is bounded if and only if it is a constant function. Thus many of the functionals in E 0 are unbounded, while for example, all of the functionals considered in [11] are bounded.…”

“…In [5], Chang et al established an interesting relationship between the integral transform and the convolution product for functionals on an abstract Wiener space. In this paper, we study the relationships that exist among the integral transform, the convolution product, and the first variation [1,4,9,11]. …”

Parts Formulas Involving Conditional Integral Transforms on Function Space

“…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.…”

Fourier-Feynman Transform and Convolution of Fourier-Type Functionals on Wiener Space

“…Various results involving Fourier-Feynman transform on Wiener space have been established and research based on this definition is continuing at the present time [1,3,4,5,12]. Recently, Kim, Kim and Yang extended the concepts of Fourier-Feynman transform and convolution on Wiener space to the concept of Fourier-Yeh-Feynman transform and convolution on Yeh-Wiener space [10,11].…”