#### Abstract

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.

#### 1. Introduction

Let denote one-parameter Wiener space, that is, the space of all real-valued continuous functions on with . Let denote the class of all Wiener measurable subsets of , and let denote Wiener measure. Then is a complete measure space, and we denote the Wiener integral of a Wiener integrable functional by

In [1], Bearman gave a significant theorem for Wiener integral on product Wiener space. It can be summarized as follows.

Theorem 1.1 (Bearman's Rotation Theorem). *Let be an -integrable functional on , the product of copies of , and let be a function of bounded variation on . Let be the transformation defined by with
**
Then the transform is measure preserving and
*

As a special case of Theorem 1.1, one can obtain the following corollary.

Corollary 1.2. *Let be Wiener integrable on . Then for any , is integrable on and
*

The following more general case of Corollary 1.2 is due to Cameron and Storvick [2]. But we state the theorem with some assumption for our research.

Theorem 1.3. *Let be Wiener measurable on . Assume that for any , is Winer integrable. Then for any , is integrable on and
*

In many papers, Theorem 1.3 is used to study relationships between analytic Fourier-Feynman transforms and convolution products of Feynman integrable functionals on Wiener space, see for instance [3–6]. In this paper, we will extend the result in Theorem 1.3 to a more general case for functionals of Gaussian processes given by (2.2) below. We then apply our rotation property of Wiener measure to establish a fundamental relationship between the generalized Fourier-Feynman transform and the generalized convolution product.

#### 2. A Rotation on Wiener Space

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 [7].

A subset of is said to be scale-invariant measurable [8] provided for all , and a scale-invariant measurable set is said to be scale-invariant null provided for all . A property that holds except on a scale-invariant null set is said to be hold scale-invariant almost everywhere (s-a.e.). If two functionals and are equal s-a.e., we write .

Let be a complete orthonormal set in , each of whose elements is of bounded variation on . Then for each , the Paley-Wiener-Zygmund (PWZ) stochastic integral is defined by the formula for all for which the limit exists, where denotes the -inner product.

It was shown in [7] that for each , the limit defining the PWZ integral exists for -a.e. and that this limit is essentially independent of the choice of the complete orthonormal set . It was also shown in [7] that if is of bounded variation on , then the PWZ integral equals the Riemann-Stieltjes integral for -a.e. . In fact, the integrals are equal for s-a.e. and that for all , is a Gaussian random variable with mean and variance .

For any with , let be the Gaussian process introduced by Park and Skoug in [9] and used extensively since; see for example [5, 6, 10, 11]. Of course if on , then .

It is easy to see that is a Gaussian process with mean zero and covariance function In addition, is stochastically continuous in on , and for any ,

For any complete orthonormal set in and for any , define the projection map from into by Then for and , we see that that is, converges in -mean to .

Throughout this paper, we will assume that each functional we consider is scale-invariant measurable and that for all .

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

Theorem 2.1. *Let be a functional on . Then for any ,
**
where , , and are related by
**
for some complete orthonormal set in , each of those elements is of bounded variation on .*

#### 3. Proof of the Main Theorem

We begin this section with three lemmas in order to establish (2.8).

Lemma 3.1. *Let be a functional on , and let be a function of bounded variation on . Then for all ,
*

*Proof. *We first note that for each ,
We also note that is Wiener integrable as a functional of . Hence, by (1.5), we obtain that for all ,
Thus (3.1) is established.

Lemma 3.2. *Let be a functional on . Then for any and each ,
**
where , , and are related by (2.9).*

*Proof. *Since the addition is continuous in the uniform topology on , we can apply (3.1) to the functional . Thus using (2.5) and (3.1), we have
Thus (3.4) is established.

Lemma 3.3. *Let be bounded and continuous on . Then for any ,
**
where , , and are related by (2.9) above.*

*Proof. *We clearly see that is Wiener integrable. We also note that is a sequence of functions of bounded variation on such that converges to in the space as . For each and , let . Since converges to uniformly and is continuous in the uniform topology, by (2.6),
Since is bounded, by using the dominated convergence theorem and (3.4), we have
which concludes the proof of Lemma 3.3.

We are now ready to prove our main theorem.

*Proof of Theorem 2.1. *Let be Wiener integrable. Suppose that the left-hand side of (2.8) exists. By usual arguments of integration theory, there exists a sequence of bounded and continuous functionals such that converges to . By Lemma 3.3 and the dominated convergence theorem, we can obtain the desired result.

Corollary 3.4. *Let be a functional on . Then for all and all ,
*

*Proof. *Simply choose and in (2.8) and use the linearity property of the PWZ stochastic integral.

Using similar arguments as in the proofs of Lemmas 3.1, 3.2, and 3.3 and Theorem 2.1 above, we can obtain the following theorems.

Theorem 3.5. *Let be a functional on , and let be any subset of . Then
**
where is the product Wiener measure on , the product of copies of , and
**
for some complete orthonormal set in .*

Theorem 3.6. *Let be a functional on . Then for any and in ,
**
where , , and are related by (2.9).*

*Remark 3.7. *Equations (2.8) and (3.12) are indeed very general formulas.(1)For any , choosing and in (2.8) yields (1.4).(2)For any , choosing and in (2.8) or choosing in (3.9) yields (1.5). (3)For any function of bounded variation , choosing and on in (3.12) yields (1.3).

#### 4. Generalized Fourier-Feynman Transform and Generalized Convolution Product

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 , with the Wiener integral and then extends analytically in to the right-half complex plane. Here we start with the (generalized) Wiener integral where is the Gaussian process given by (2.2) above.

Throughout this section, let and denote the complex numbers with positive real part and the nonzero complex numbers with nonnegative real part, respectively.

Let be a complex-valued scale-invariant measurable functional on such that given by (4.2) exists and is finite for all . If there exists a function analytic on such that for all , then is defined to be the generalized analytic Wiener integral (with respect to the process ) of over with parameter , and for we write Let be a nonzero real number and let be a functional such that exists for all . If the following limit exists, we call it the generalized analytic Feynman integral of with parameter and we write where approaches through values in .

Note that if on , then these definitions agree with the previous definitions of the analytic Wiener integral and the analytic Feynman integral [3, 4, 8, 12–14].

Next (see [5, 6, 15]) we state the definition of the generalized Fourier-Feynman transform (GFFT).

*Definition 4.1. *For and , let
Let be a non-zero real number. For , we define the analytic GFFT with respect to , of , by the formula ,
if it exists; that is, for each ,
where . We define the analytic GFFT, of , by the formula
if it exists.

We note that for , is defined only s-a.e. We also note that if exists and if , then exists and . One can see that for each , since

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

*Definition 4.2. *Let and be scale-invariant measurable functionals on . For and , we define their GCP with respect to (if it exists) by
When , we denote by .

*Remark 4.3. *Our definition of the GCP is different than the definition given by Huffman et al. in [5, 6] and used by Chang et al. in [15]. But if we choose in (4.10), our GCP is the GCP used in [5, 6, 15].

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

Lemma 4.4. *Let be a subset of , and let be given by
**
respectively. Then the following assertions are equivalent. *(i)* and are independent processes.*(ii)*. *

*Proof. *Since the processes and are Gaussian with mean zero, we know that and are independent processes if and only if
for every . But, using (2.4), we have
From this, we can obtain the desired result.

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

Lemma 4.5. *Let and be functionals on . Let be a subset of such that almost everywhere on , and let
**
Furthermore, assume that for all , , and all exist. Then
**
for s-a.e. .*

*Proof. *We note that for all ,
But , and so and are independent processes by Lemma 4.4. Hence by (2.8), we obtain that for all ,
Equation (4.15) holds for all by analytic continuation.

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

Theorem 4.6. *Let , , , , and be as in Lemma 4.5. Furthermore, assume that for , and , , , , and all exist and that
**
Then
**
for s-a.e. .*

*Proof. *Equation (4.19) follows from (4.15) by letting , since all transforms in (4.18) and (4.19) exist.

*Remark 4.7. *We note that the hypotheses (and hence the conclusions) of Theorem 4.6 above are indeed satisfied by many of the functionals in the following large classes of functionals. These classes of functionals include; (i) the Banach algebra defined by Cameron and Storvick in [16]: also see [3, 5, 14, 15],(ii) various spaces of functionals of the form
for appropriate as discussed in [4, 12, 13]; and(iii) various spaces of functionals of the form
for appropriate as discussed in [3].

Next five corollaries include the results of [3–6] by Huffman et al. The notations used in [3–6] are slightly different than ours.

Corollary 4.8. *Refer to Theorem 2.1 in [5].*

*Proof. *In our Lemma 4.5, simply choose .

Corollary 4.9. *Refer to Theorem 1 in [6].*

*Proof. *In our Theorem 4.6, simply choose .

Corollary 4.10. *Refer to Theorem 3.3 in [5].*

*Proof. *In our Theorem 4.6, simply choose .

Corollary 4.11. *Refer to Theorem 3.3 in [3].*

*Proof. *In our Theorem 4.6, simply choose .

Corollary 4.12. *Refer to Lemma 4.1 and Theorems 4.1 and 4.2 in [4].*

*Proof. *In our Lemma 4.5 and Theorem 4.6, simply choose .

#### Acknowledgment

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