Shifting and Variational Properties for Fourier-Feynman Transform and Convolution

Abstract

Shifting, scaling, modulation, and variational properties for Fourier-Feynman transform of functionals in a Banach algebra are given. Cameron and Storvick's translation theorem can be obtained as a corollary of our result. We also study shifting, scaling, and modulation properties for the convolution product of functionals in .

1. Introduction

Let denote the Wiener space, that is, the space of real valued continuous functions on with . The concept of analytic Fourier-Feynman transform for functionals on Wiener space was introduced by Brue in [1]. In [2], Cameron and Storvick introduced analytic Fourier-Feynman transform. In [3], Johnson and Skoug developed analytic Fourier-Feynman transform for that extended the results in [2].

In [4, 5], Huffman et al. defined a convolution product for functionals on Wiener space and showed that the Fourier-Feynman transform of a convolution product is a product of Fourier-Feynman transforms. Recently Kim et al. [6] obtained change of scale formulas for Wiener integrals related to Fourier-Feynman transform and convolution. For a detailed survey of the previous work on Fourier-Feynman transform and related topics, see [7].

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 [8] 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 (s-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 with in this paper. The Banach algebra , which was introduced by Cameron and Storvick [9], consists of functionals expressible in the form for s-a.e. in , where the associated measure is a complex Borel measure on and denote the Paley-Wiener-Zygmund stochastic integral .

In this paper, we study shifting, scaling, modulation, and variational properties for Fourier-Feynman transform of functionals in . Shifting properties are some of the important properties of Fourier transform. In Section 2, we develop shifting properties for Fourier-Feynman transform. For example, time shifting, frequency shifting, scaling, and modulation properties for Fourier-Feynman transform are given.

In Section 3, we study variational properties for Fourier-Feynman transform of functionals in and in the last section we develop shifting, scaling, and modulation properties for convolution product of functionals in .

The Banach algebra is a very rich class of functionals. It is known that important functionals in quantum mechanics and Feynman integration theory belong to . For example, functionals of the form were discussed in the book by Feynman and Hibbs [10] on path integrals and in Feynman’s original paper [11]. For appropriate , functionals of form (6) are known to belong to [12]. Hence the results in this paper can be immediately applied to many functionals of form (6).

2. Shifting Properties for Fourier-Feynman Transform

In this section we develop some of the important properties relevant to shifting (translating) and computational rules for Fourier-Feynman transform of functionals in the Banach algebra . Let us begin with the definition of the Fourier-Feynman transform of functionals on Wiener space.

Let and let be a nonzero real number throughout this paper.

Definition 1. Let be a functional on . For and , let For , we define analytic Fourier-Feynman transform of on by the formula whenever this limit exists; that is, for each , where . We define analytic Fourier-Feynman transform of by for s-a.e. , whenever this limit exists [2–5].

Since is linear, obviously is linear; that is, for all constants , and functionals , on , whenever each transform exists.

By Definition (4) of the analytic Feynman integral and analytic Fourier-Feynman transform (10), it is easy to see that In particular, if , then is analytic Feynman integrable and

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

Theorem 2 (Theorem 3.1 of [5]). Let be given by (5). Then for all for s-a.e. . Moreover the Fourier-Feynman transform exists, belongs to , and is given by for s-a.e. .

The Fourier transform turns a function into a new function . Because the transform is used in signal analysis, we usually use for time as the variable with and as the variable of the transform ; that is, Engineers refer to the variable in the transformed function as the frequency of the signal [13].

We will use the same convention in this paper; that is, for a Fourier-Feynman transform of , we call the variable a time and the variable a frequency.

Our first result in this section shows that the time shifting of the Fourier-Feynman transform is equal to the frequency shifting of the Fourier-Feynman transform.

Theorem 3. Let be a functional on and let . Then one has if all sides exist.

Proof. For all and for s-a.e. , if the Wiener integral exists. Hence we have the result.

The following theorem is reminiscent of the time shifting theorem for the Fourier transform. Hence we call the following theorem the time shifting formula for Fourier-Feynman transform on Wiener space. It says that if we shift back and replace by , then the Fourier-Feynman transform of this shifted function is equal to the Fourier-Feynman transform of multiplied by an exponential factor.

Theorem 4 (time shifting). Let be given by (5) and let . Then one has for s-a.e. .

Proof. Let . Using (5) we write as where for a Borel subset of ; that is, also belongs to . Now by Theorem 2, we have Finally by Theorem 2 again we have and so by (17) the proof is complete.

Cameron and Storvick [14] presented a new translation theorem for the analytic Feynman integral on Wiener space. Moreover Ahn et al. [15] gave a simple proof of an abstract Wiener space version of the translation theorem. Taking and in (19) and considering (13) we obtain Cameron and Storvick’s translation theorem as follows. Hence Theorem 4 can be viewed as Cameron and Storvick’s translation theorem for the Fourier-Feynman transform.

Corollary 5. Let be given by (5) and let . Then one has

Next theorem is reminiscent of the frequency shifting theorem for the Fourier transform. Using Theorem 3 we have the following property for the frequency shifting of the Fourier-Feynman transform.

Theorem 6 (frequency shifting). Let be given by (5) and let . Then one has for s-a.e. .

The following theorem is called a scaling theorem because we want the transform not of , but of , in which can be thought of as a scaling factor.

Theorem 7 (scaling). Let be given by (5) and let be a nonzero real number. Then one has for s-a.e. .

Proof. Let for . Using (5) we can write as where for each Borel subset of . By Theorem 2, we have for s-a.e. . Finally by Theorem 2 again we obtain the result.

Our next corollary follows immediately from the scaling theorem above by putting . This result is called time reversal because we replace by in to get . The transform of this new functional is obtained by simply replacing by in the transform of .

Corollary 8 (time reversal). Let be given by (5). Then one has for s-a.e. .

Our next theorem is useful in obtaining the Fourier-Feynman transforms of new functionals from the Fourier-Feynman transforms of old functionals for which we know their Fourier-Feynman transform.

Theorem 9 (modulation). Let be given by (5) and let . Then one has where for s-a.e. .

Proof. Put and use the linearity of the Fourier-Feynman transform to get Finally by the time shifting theorem or frequency shifting theorem we obtain (29). Using the second conclusion is proved similarly.

Since the Dirac measure concentrated at in is a complex Borel measure, the constant function belongs to . Hence we have the following corollary.

Corollary 10. Let . Then one has for s-a.e. .

Proof. Since for , by the modulation property, Theorem 9, and Euler’s formula, (33) and (34) follow immediately.

3. Variational Properties for Fourier-Feynman Transform

In using the Fourier transform to solve differential equations, we need an expression relating the transform of to that of . In this section we develop similar relationships for Fourier-Feynman transform on Wiener space; that is, we provide variational properties for Fourier-Feynman transform of functionals in the Banach algebra .

Definition 11. Let be a functional on and let . Then (if it exists) is called the first variation of . The higher order variations of are defined inductively. For example, the 2nd order variation of is the first variation of with respect to and is defined by and th order variation is defined by for . If , then we denote as .

Theorem 12. Let be given by (5) with . Then for s-a.e. and in , exists, is an element of as a function of , and is given by the formula where is a complex Borel measure on defined by for each Borel subset of .

Proof. We will prove that for s-a.e. and , in . Using Lemma 3.1 of [16], we have Now the result follows if we can pass the differentiation under the integral sign. But this is done because by the Fubini theorem which is finite and so for s-a.e. , in . Now by mathematical induction we obtain general result (38).

If in Theorem 12, then we have the following corollary.

Corollary 13. Let be given by (5) with . Then for s-a.e. and in , exists, is an element of as a function of , and is given by the formula where is the complex Borel measure on defined by for each Borel subset of .

In our next theorem, for functionals in we establish a relationship between the Fourier-Feynman transform of the variation and the variation of the Fourier-Feynman transform. Also see Corollary 4.3 of [15] for a similar result.

Theorem 14. Let be given by (5) with . Then one has for s-a.e. and in . Also, both of the expressions in (45) are given by the expression for s-a.e. and in .

Proof. By (38) and Theorem 2, we have for s-a.e. and in . Now by (39) we know that the last expression can be rewritten as (46). On the other hand, by the same method as in the proof of Theorem 12, we see that the right hand side of (45) is also expressed as (46).

Letting in Theorem 14 we have the following corollary.

Corollary 15. Let be given by (5) with . Then one has for s-a.e. and in .

The following theorem involves an iterated Fourier-Feynman transform of th order variation.

Theorem 16. Let be given by (5) with . Then one has for s-a.e. and in .

Proof. Let . For , we obtain for s-a.e. and in . Since , the Fubini theorem enables us to conclude that Since the right hand side of the last expression is independent of , we have for s-a.e. and in .
Now considering Theorem 12 and applying repeatedly the first part of this proof, we obtain for s-a.e. and in . Finally by Theorem 12 again the proof is complete.

4. Shifting Properties for Convolution Product

We developed in Section 2 some properties relevant to shifting and computational rules for the Fourier-Feynman transform of functionals in the Banach algebra . In this section we study similar properties for the convolution product of functionals in . Let us begin with the definition of the convolution product of functionals on Wiener space.

Definition 17. Let and be functionals on . For and , one defines the convolution product (if it exists) by

Obviously the convolution is bilinear in the sense that for all functionals , on for , whenever each convolution exists.

Huffman et al. established the existence of the convolution product on for functionals in .

Theorem 18 (Theorem 3.2 of [5]). Let and be elements of with corresponding finite Borel measures and , respectively. Then their convolution product exists and is given by the formula for s-a.e. .

Our first result in this section is a relationship between time shifting and frequency shifting of the convolution product on Wiener space.

Theorem 19. Let and be functionals on and let . Then one has if each side exists.

Proof. For all and for s-a.e. , we have if the Wiener integrals exist. Hence we have the result.

The following theorem is reminiscent of the time shifting theorem for Fourier-Feynman transform (Theorem 4) in Section 2. But in this theorem we have to shift back for and shift front for to obtain a concrete form of a time shifting formula for the convolution product.

Theorem 20 (time shifting). Let and be given as in Theorem 18 and let . Then one has for s-a.e. .

Proof. Let and . By (5) we have where and for a Borel subset of . Then, by Theorem 18, the left hand side of (60) is given by for s-a.e. .
To consider the right hand side of (60), let where and for a Borel subset of . Then, by Theorem 18, we have for s-a.e. . Since , we have for s-a.e. and this completes the proof.

Considering the second part of the proof of Theorem 20 above, we see that, for and given as in Theorem 18, for s-a.e. .

The following is a scaling theorem for the convolution product.

Theorem 21 (scaling). Let and be given as in Theorem 18. If is a nonzero real number, then for s-a.e. .

Proof. By the same method as used in the proof of Theorem 7 we have for s-a.e. . Hence by Theorem 18 we obtain the result.

Our next corollary follows immediately from the scaling theorem above by putting . This result is called time reversal because we replace by in and to get and , respectively. The convolution of these new functionals is obtained by simply replacing by in the convolution of and .

Corollary 22 (time reversal). Let and be given as in Theorem 18. Then one has for s-a.e. .

Our next theorem is useful to obtain the convolution product of new functionals from the convolution product of old functionals when we know their convolution product.

Theorem 23 (modulation). Let and be given as in Theorem 18 and let . Then where for s-a.e. .

Proof. Put and use bilinearity (56) of the convolution product to get Finally by (60) and (68) we obtain (72). Using the other conclusions are proved similarly.

Since the Dirac measure concentrated at in is a complex Borel measure, the constant function belongs to . Hence we have the following corollary.

Corollary 24. Let . Then one has for s-a.e. .

Proof. Since for , by the modulation property, Theorem 23, and Euler’s formula, the results follow immediately.

Conflict of Interests

The author declares 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).