We first investigate a rotation property of Wiener measure on the product of Wiener spaces. Next, using the concept of the generalized analytic Feynman integral, we define a generalized Fourier-Feynman transform and a generalized convolution product for functionals on Wiener space. We then proceed to establish a fundamental result involving the generalized transform and the generalized convolution product.

Let

In [

Let

As a special case of Theorem

Let

The following more general case of Corollary

Let

In many papers, Theorem

The most important concepts we will employ in the statements and proofs of our results are the concepts of the scale-invariant measurability and the Paley-Wiener-Zygmund stochastic integral [

A subset

Let

It was shown in [

For any

It is easy to see that

For any complete orthonormal set

Throughout this paper, we will assume that each functional

We are now ready to state the main theorem of this paper.

Let

We begin this section with three lemmas in order to establish (

Let

We first note that for each

Let

Since the addition is continuous in the uniform topology on

Let

We clearly see that

We are now ready to prove our main theorem.

Let

Let

Simply choose

Using similar arguments as in the proofs of Lemmas

Let

Let

Equations (

For any

For any

For any function of bounded variation

In this section, we will apply our main theorem to the generalized analytic Fourier-Feynman transform and the convolution product theories.

In defining various analytic Feynman integrals, one usually starts, for

Throughout this section, let

Let

Note that if

Next (see [

For

We note that for

Next we give the definition of the generalized convolution product (GCP).

Let

Our definition of the GCP is different than the definition given by Huffman et al. in [

We begin this section with a key lemma for a relationship between the GFFT and the GCP.

Let

Since the processes

We are now ready to establish fundamental relationships between the GFFT and the GCP.

Let

We note that for all

In next theorem, we show that the GFFT of the GCP is the product of GFFTs.

Let

Equation (

We note that the hypotheses (and hence the conclusions) of Theorem

the Banach algebra

various spaces of functionals of the form

various spaces of functionals of the form

Next five corollaries include the results of [

Refer to Theorem 2.1 in [

In our Lemma

Refer to Theorem 1 in [

In our Theorem

Refer to Theorem 3.3 in [

In our Theorem

Refer to Theorem 3.3 in [

In our Theorem

Refer to Lemma 4.1 and Theorems 4.1 and 4.2 in [

In our Lemma

The present research was conducted by the research fund of Dankook University in 2010.