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We investigate the partial derivative approach to the change of scale formula for the functon space integral and we investigate the vector calculus approach to the directional derivative on the function space and prove relationships among the Wiener integral and the Feynman integral about the directional derivative of a Fourier transform.

We shall prove the existence of the Wiener integral and the analytic Wiener and Feynman integral and we obtain those relationships and later, we prove the change of scale formula for the Wiener integral about the first variation of a function defined on the product abstract Wiener space. Later, we obtain those relationships in the Fresnel class as it’s corollaries.

Let C[0, t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0, t], and define a random vector Z : C[0, t] → R by $${Z_n}\left( x \right) = \left( {x\left( 0 \right) + a\left( 0 \right),\int_o^{{t_1}} {h\left( s \right)dx\left( s \right) + x\left( 0 \right) + a\left( {{t_1}} \right),...,\int_0^{{t_n}} {h\left( s \right)dx\left( s \right) + x\left( 0 \right) + a\left( {{t_n}} \right)} } } \right)$$ , where a ∈ C[0, t], h ∈ L [0, t], and 0 < t <... < t ≤ t is a partition of [0, t]. Using simple formulas for generalized conditional Wiener integrals, given Z we will evaluate the generalized analytic conditional Wiener and Feynman integrals of the functions F in a Banach algebra which corresponds to Cameron-Storvick's Banach algebra S. Finally, we express the generalized analytic conditional Feynman integral of F as a limit of the non-conditional generalized Wiener integral of a polygonal function using a change of scale transformation for which a normal density is the kernel. This result extends the existing change of scale formulas on the classical Wiener space, abstract Wiener space and the analogue of the Wiener space C[0, t]. [ABSTRACT FROM AUTHOR]

Cameron and Storvick discovered change of scale formulas for Wiener integrals of functionals in a Banach algebra S on classical Wiener space. Yoo and Skoug extended these results for functionals in the Fresnel class F(B) and in a generalized Fresnel class FA1,A2 on abstract Wiener space. We establish a relationship between a function space integral and a generalized analytic Feynman integral on Ca,b[0,T] for functionals in a Banach algebra S(La,b²[0, T]). Moreover, we obtain a change of scale formula for a function space integral on Ca,b[0,T] of these functionals. [ABSTRACT FROM AUTHOR]

We establish the analytic Wiener and Feynman integral theory for Wiener integrals over paths on C 0 ( B ), the space of abstract Wiener space valued continuous functions on [0, T ], for certain cylinder-type functions of the form: where f ??? L p (R mn ), 1?=? p ?=?8, and 0?=? s 0 ?=? s 1 ?=?&CenterDot;?&CenterDot;?&CenterDot;?=? s n ?=? T is a partition of [0, T ]. In addition, we establish some relationships between analytic Feynman integrals, Fourier–Feynman transforms and Wiener integrals and we prove the change of scale formula for Wiener integrals over paths on C 0 ( B ) in abstract Wiener spaces. [ABSTRACT FROM AUTHOR]

We shall prove the existence of the Wiener integral and the analytic Wiener and Feynman integral and we obtain those relationships and later, we prove the change of scale formula for the Wiener integral about the first variation of a function defined on the product abstract Wiener space. Later, we obtain those relationships in the Fresnel class as it's corollaries. [ABSTRACT FROM AUTHOR]

We investigate the partial derivative approach to the change of scale formula for the functon space integral and we investigate the vector calculus approach to the directional derivative on the function space and prove relationships among the Wiener integral and the Feynman integral about the directional derivative of a Fourier transform. [ABSTRACT FROM AUTHOR]

We investigate the partial derivative approach to the change of scale formula for the functon space integral and we investigate the vector calculus approach to the directional derivative on the function space and prove relationships among the Wiener integral and the Feynman integral about the directional derivative of a Fourier transform.