Journal of Function Spaces

Volume 2014 (2014), Article ID 657934, 7 pages

http://dx.doi.org/10.1155/2014/657934

## Change of Scale Formulas for Wiener Integrals Related to Fourier-Feynman Transform and Convolution

^{1}Department of Mathematics, Daejin University, Pocheon 487-711, Republic of Korea^{2}School of Liberal Arts, Seoul National University of Science and Technology, Seoul 139-743, Republic of Korea^{3}Department of Mathematics, Yonsei University, Wonju 220-710, Republic of Korea

Received 18 April 2014; Accepted 19 May 2014; Published 26 June 2014

Academic Editor: Ismat Beg

Copyright © 2014 Bong Jin Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Cameron and Storvick discovered change of scale formulas for Wiener integrals of functionals in Banach algebra on classical Wiener space. Yoo and Skoug extended these results for functionals in the Fresnel class and in a generalized Fresnel class on abstract Wiener space. We express Fourier-Feynman transform and convolution product of functionals in as limits of Wiener integrals. Moreover we obtain change of scale formulas for Wiener integrals related to Fourier-Feynman transform and convolution product of these functionals.

#### 1. Introduction

It has long been known that Wiener measure and Wiener measurability behave badly under the change of scale transformation [1] and under translations [2]. Cameron and Storvick [3] expressed the analytic Feynman integral on classical Wiener space as a limit of Wiener integrals. In doing so, they discovered nice change of scale formulas for Wiener integrals on classical Wiener space [4]. In [5, 6], Yoo and Skoug extended these results to an abstract Wiener space . Moreover Yoo et al. [7, 8] established a change of scale formula for Wiener integrals of some unbounded functionals on (a product) abstract Wiener space. Recently Yoo et al. [9] obtained a change of scale formula for a function space integral on a generalized Wiener space .

On the other hand, in [10], Cameron and Storvick introduced an analytic Fourier-Feynman transform. In [11], Johnson and Skoug developed an analytic Fourier-Feynman transform for that extended the results in [10]. In [12], Huffman et al. defined a convolution product for functionals on Wiener space and, for a cylinder type functional, showed that the Fourier-Feynman transform of the convolution product is a product of Fourier-Feynman transforms. For a detailed survey of the previous work on the Fourier-Feynman transform and related topics, see [13].

In this paper, we express the Fourier-Feynman transform and convolution product of functionals in Banach algebra as limits of Wiener integrals on . Moreover we obtain change of scale formulas for Wiener integrals related to Fourier-Feynman transform and convolution product of these functionals. Some preliminary results of this paper were introduced as an oral presentation in 2013 Annual Meeting of the Korean Mathematical Society [14].

Let denote the Wiener space, that is, the space of 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 functional by

A subset of is said to be scale-invariant measurable [15] provided is measurable for each , and a scale-invariant measurable set is said to be scale-invariant null provided for each . A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (-a.e.).

Let and denote the sets of complex numbers with positive real part and the complex numbers with nonnegative real part, respectively. Let be a complex valued measurable functional on such that the Wiener integral exists as a finite number for all . If there exists a function analytic in such that for all , then is defined to be the analytic Wiener integral of over with parameter , and for we write If the following limit exists for nonzero real , then we call it the analytic Feynman integral of over with parameter and we write where approaches through .

Now we will introduce the class of functionals that we work on in this paper. The Banach algebra was introduced in [16] by Cameron and Storvick. It consists of functionals expressible in the form for -a.e. in , where the associated measure is a complex Borel measure on and denote the Paley-Wiener-Zygmund stochastic integral .

#### 2. Fourier Feynman Transform and a Change of Scale Formula

In this section we give a relationship between the Wiener integral and the Fourier-Feynman transform on for functionals in the Banach algebra ; that is, we express the Fourier-Feynman transform of functionals in as a limit of Wiener integrals on . We begin this section by introducing the definition of analytic Fourier-Feynman transform for functionals defined on . Let and let be a nonzero real number.

*Definition 1. *Let be a functional on . For and , let
For , we define the analytic Fourier-Feynman transform of on by the formula
whenever this limit exists; that is, for each ,
where . We define the analytic Fourier-Feynman transform of by
for -a.e. , whenever this limit exists [10–12, 17].

By the definition (4) of the analytic Feynman integral and the analytic Fourier-Feynman transform (9), it is easy to see that for a nonzero real number , In particular, if , then is analytic Feynman integrable and

Huffman et al. established the existence of the Fourier-Feynman transform on for functionals in .

Theorem 2 (Theorem 3.1 of [17]). *Let be given by (5). Then for all **
for -a.e. . Moreover for all and for all nonzero real number , the Fourier-Feynman transform exists and is given by
**
for -a.e. .*

We next introduce an integration formula which is useful in this paper. The proof of this lemma is essentially the same as Lemma 3 of [3] and hence we will state it without proof.

Lemma 3. *Let , let be an orthonormal set in , and let . Then
*

Now we give a relationship between the analytic Fourier-Feynman transform and the Wiener integral on for functionals in . In this theorem we express the Fourier-Feynman transform of functionals in as a limit of Wiener integrals.

Theorem 4. *Let be given by (5). Let be a complete orthonormal set of functionals in . Let be a nonzero real number and let be a sequence of complex numbers in such that . Then we have
**
for -a.e. .*

*Proof. *Let be the Wiener integral on the right hand side of (15). Then by (5) and the Fubini theorem,
where
By Lemma 3, we evaluate the above Wiener integral to obtain
Now by Parseval’s theorem
and so by the bounded convergence theorem
Finally by (13) in Theorem 2, the proof is completed.

As we have seen in (10) and (11) above, if , then the Fourier-Feynman transform is equal to the analytic Feynman integral of . Hence we have the following corollary.

Corollary 5 (Theorem 2 of [3]). *Let be given by (5). Let be a complete orthonormal set of functionals in . Let be a nonzero real number and let be a sequence of complex numbers in such that . Then we have
*

The following is a relationship between and the Wiener integral for functionals in .

Theorem 6. *Let be given by (5). Let be a complete orthonormal set of functionals in . Then for each we have
**
for -a.e. .*

*Proof. *To prove this theorem, we modify the proof of Theorem 4 by replacing by whenever it occurs. Then we have
We apply the dominated convergence theorem to obtain
By (12) in Theorem 2, the proof is completed.

Our main result in this section, namely, a change of scale formula for Wiener integral related to Fourier-Feynman transform of functionals in , now follows from Theorem 6.

Theorem 7. *Let be given by (5). Let be a complete orthonormal set of functionals in . Then for each **
for -a.e. .*

*Proof. *First note that for
Letting in (22), we have (25) and this completes the proof.

Letting in (25), we have the following change of scale formula for Wiener integrals on classical Wiener space.

Corollary 8 (Theorem 2 of [4]). *Let be given by (5). Let be a complete orthonormal set of functionals in . Then we have
**
for each .*

In our next example we will explicitly compute a Wiener integral of a functional under a change of scale transformation.

*Example 9. *Let be a complete orthonormal set of functionals in . Define
for and is a real or complex number. We evaluate the Wiener integrals on each side of (25). The left hand side of (25) can be evaluated as follows:
By the Paley-Wiener-Zygmund theorem (see [18]), we have
Next we evaluate the Wiener integral on the right hand side of (25). Consider
By the Paley-Wiener-Zygmund theorem again, we have
Thus we have established that (25) is valid for .

Note that in Example 9 above, is a real or complex number. If is pure imaginary, and is an example of a functional to which Theorem 7 applies. On the other hand, if the real part of is not equal to , then can be unbounded. Thus this example shows that the class of functionals for which (25) holds is more extensive than .

#### 3. Convolution and a Change of Scale Formula

In this section we give a relationship between the Wiener integral and the convolution product on for functionals in the Banach algebra ; that is, we express the convolution product of functionals in as a limit of Wiener integrals on . We start this section by introducing the definition of convolution product for functionals on .

*Definition 10. *Let and be functionals on . For and , we define their convolution product by
if it exists. Moreover for nonzero real number , the convolution product is defined by
if it exists [12, 17, 19, 20].

The following is the existence theorem for the convolution product of functionals in on .

Theorem 11 (Theorem 3.2 of [17]). *Let and be elements of with associated complex Borel measures and , respectively. Then for all **
for -a.e. . Moreover for all nonzero real number , the convolution product exists and is given by
**
for -a.e. .*

Now we give a relationship between the convolution product and the Wiener integral on for functionals in . In this theorem we express the convolution product of functionals in as a limit of Wiener integrals.

Theorem 12. *Let and be elements of with associated complex Borel measures and , respectively. Let be a complete orthonormal set of functionals in . Let be a nonzero real number and let be a sequence of complex numbers in such that . Then we have
**
for -a.e. .*

*Proof. *Let be the Wiener integral on the right hand side of (37). Then by (5) and the Fubini theorem,
where
By Lemma 3, we evaluate the above Wiener integral to obtain
Now by Parseval’s theorem
and so by the bounded convergence theorem
Finally by (36) in Theorem 11, the proof is completed.

The following is a relationship between the convolution product and the Wiener integral for functionals in .

Theorem 13. *Let and be elements of with associated complex Borel measures and , respectively. Let be a complete orthonormal set of functionals in . Then for each we have
**
for -a.e. .*

*Proof. *To prove this theorem, we modify the proof of Theorem 12 by replacing by whenever it occurs. Then we have
We apply the dominated convergence theorem to obtain
By (35) in Theorem 11, the proof is completed.

Our main result in this section, namely, a change of scale formula for Wiener integral related to convolution product of functionals in , now follows from Theorem 13.

Theorem 14. *Let and be elements of with associated complex Borel measures and , respectively. Let be a complete orthonormal set of functionals in . Then for each **
for -a.e. .*

*Proof. *First note that for
Letting in (43), we have (46) and this completes the proof.

In our final example we will explicitly compute a Wiener integral related to convolution product under a change of scale transformation.

*Example 15. *Let be a complete orthonormal set of functionals in . Define
for and is a real or complex number. We evaluate the Wiener integrals on each side of (46). The left hand side of (46) can be evaluated as follows:
By the Paley-Wiener-Zygmund theorem, we have
Next we evaluate the Wiener integral on the right hand side of (46):
By the Paley-Wiener-Zygmund theorem, we have
Thus we have established that (46) is valid for and .

Note that in Example 15 above, was a real or complex number. If is pure imaginary, and belong to , so and are examples of functionals to which Theorem 14 applies. On the other hand, if the real part of is not equal to , then and can be unbounded. Thus this example shows that the class of functionals for which (46) holds is more extensive than .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0022563).

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