Abstract and Applied Analysis

Volume 2013 (2013), Article ID 565832, 12 pages

http://dx.doi.org/10.1155/2013/565832

Research Article

Sequential Generalized Transforms on Function Space

Jae Gil Choi, Hyun Soo Chung, and Seung Jun Chang

Department of Mathematics, Dankook University, Cheonan 330-714, Republic of Korea

Received 20 August 2013; Accepted 2 November 2013

Academic Editor: Douglas Anderson

Copyright © 2013 Jae Gil Choi 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.

We define two sequential transforms on a function space induced by generalized Brownian motion process. We then establish the existence of the sequential transforms for functionals in a Banach algebra of functionals on . We also establish that any one of these transforms acts like an inverse transform of the other transform. Finally, we give some remarks about certain relations between our sequential transforms and other well-known transforms on .

1. Introduction and Preliminaries

Let denote one-parameter Wiener space; that is, the space of all real-valued continuous functions on with . The study of the Fourier-Wiener transform of functionals on was initiated by Cameron and Martin [1–3]. This transform and its properties are similar in many respects to the ordinary Fourier function transform. Since then, many transforms which were somewhat analogous to the Fourier-Wiener transform have been defined and developed in the literature. There are two well-known transforms on . One of them is the analytic Fourier-Feynman transform [4–6] and the other is the integral transform [7–10]. Each of the transforms on Wiener space has an inverse transform. For an elementary survey, see [11].

In [12–16], the authors studied the generalized analytic Fourier-Feynman transform and the generalized integral transform for functionals defined on a more general function space . The function space , induced by generalized Brownian motion process, was introduced by Yeh [17, 18] and was used extensively by Chang and Chung [19]. The Wiener process used in [1–10] is stationary in time and is free of drift, while the stochastic process used in this paper, as well as in [12–17, 19], is nonstationary in time and is subject to a drift . In case and on , the general function space reduces to the Wiener space and so most of the results in [4–6, 9] follow immediately from the results in [12, 13, 15, 16].

However, the existence of an inverse transform of each of the two generalized transforms on has not yet been established. It is a critical point that the generalized transforms on are essentially different from the transforms on Wiener space . The main purpose of this paper is to define a transform on which has an inverse transform.

In this paper, we define two sequential transforms on the function space . To do this, we investigate a representation for sample paths of the generalized Brownian motion process and introduce the concept of the -s-continuity for functionals on . We then proceed to establish the existence of the sequential transforms for functionals in a Banach algebra of functionals on . Next, we establish that any one of these transforms acts like an inverse transform of the other transform. Finally, we examine certain aspects of the generalized analytic Fourier-Feynman transform, the generalized integral transform, and the sequential transforms.

We briefly list some of the preliminaries from [12, 13, 17] that we will need in order to establish the results in this paper.

Let be an absolutely continuous real-valued function on with , , and let be a strictly increasing, continuously differentiable real-valued function with and for each . The generalized Brownian motion process determined by and is a Gaussian process with mean function and covariance function . For more details, see [12, 13]. By Theorem  14.2 in [18], the probability measure induced by , taking a separable version, is supported by (which is equivalent to the Banach space of continuous functions on with under the sup norm). Hence, is the function space induced by where is the Borel -algebra of . We then complete this function space to obtain , where is the set of all Wiener measurable subsets of .

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

Let be the space of functions on which are Lebesgue measurable and square integrable with respect to the Lebesgue-Stieltjes measures on induced by and ; that is,

where is the total variation function of . Then, is a separable Hilbert space with inner product defined by

Note that if and only if a.e. on . Also note that all functions of bounded variation on are elements of . If and , then . In fact,

because the norms and are equivalent.

Let

For , with for , let be defined by the formula

Then with inner product

is a separable Hilbert space.

Note that the two separable Hilbert spaces and are homeomorphic under the linear operator given by (5). The inverse operator of is given by

Recall that above, as well as in papers [12–16], we require that is an absolutely continuous function with and with . Our conditions on imply that for some positive real numbers and , and is continuous on . Hence, we have

But we cannot ensure that

under current conditions. Note that the function , , does not satisfy condition (9) even though its derivative is an element of .

In this paper, we add the requirement (9). Then we obtain the following lemma.

Lemma 1. The function satisfies the requirement (9) if and only if is an element of .

Under the requirement (9), we observe that for each ,

where for .

The following integration formula is used several times in this paper:

for complex numbers and with .

2. A Representation for Paths in

In this section, we investigate a representation for paths in . To do this, we first define a Paley-Wiener-Zygmund (PWZ) type stochastic integral.

Let be a complete orthonormal set of functions in the separable Hilbert space , such that for each , is of bounded variation on . Then, for each with , we can write

on .

For each , the PWZ type stochastic integral is defined by the formula

for all for which the limit exists.

The following are some basic properties of the PWZ type stochastic integral. They are nontrivial, but straightforward to prove.

(1)For each , the PWZ type stochastic integral exists for -a.e. .(2)The PWZ type stochastic integral is essentially independent of the choice of the complete orthonormal set in .(3)It follows from the definition of the PWZ type stochastic integral and from Parseval’s equality that if and , then exists and we have .(4)If is of bounded variation on , then the PWZ type stochastic integral equals the Riemann-Stieltjes integral for -a.e. .(5)The PWZ type stochastic integral has the expected linearity properties. That is, for any real number , and , we have

 Thus, the assertions (1) and (4) hold for s-a.e. .(6)For each , is a Gaussian random variable with mean and variance . For all , we have

Now, we are ready to examine a representation for paths in . Throughout the rest of this paper, we will use the symbol for a complete orthonormal set in , such that for each , is of bounded variation on . Then, for each ,

is a Gaussian random variable with mean

and variance

We note that the set forms a set of independent Gaussian random variables on .

Let be as above, and let . For each , define

Then, is an element of for all and all .

For each , let . Then, we observe that for each ,

By the property of the PWZ type stochastic integral, the last expression of (20) converges for -a.e. .

Remark 2. By the definition of the PWZ type stochastic integral, the last expression of (20) is independent of the choice of the complete orthonormal set in . If we choose the complete orthonormal sine sequence in , where

that is,

for , then the corresponding converges to uniformly in with probability one. For more details, see [20] and the references therein.

We now state a fundamental integration formula on the function space . Let and be as above, let be Lebesgue measurable, and let . Then

in the sense that if either side of (23) exists, both sides exist and equality holds.

Let be a functional on and let be a complete orthonormal set in . Then, we say that is -continuous at if

where is given by (19).

Example 3. For each , let be given by . Then, using (20) and (14), we see that is -continuous for s-a.e. and all .

Example 4. Given , let be given by . Since for every , by the definition of PWZ type stochastic integral, we obtain that

for s-a.e. and all .

Proposition 5. Let be -integrable. Assume that is -continuous for -a.e. . Then

where is given by (19).

Proof. For each , let . Then is -integrable. By our assumption, we observe that

for -a.e. . Thus, by the dominated convergence theorem, it follows that

as desired.

3. Sequential Function Space Integrals

In [21], Cameron and Storvick defined the “sequential” Feynman integral by means of finite dimensional approximations for functionals on Wiener space . The sequential definition for the Feynman integral was intended to interpret the Feynman’s uniform measure [22] on path space , because there is no countably additive measure which weights all paths in equally in much the same way as Lebesgue measure weights all points in equally. Thus, the Cameron and Storvick’s sequential Feynman integral is a rigorous mathematical formulation of the Feynman’s path integral.

The Cameron and Storvick’s sequential Feynman integral is related by sequential Wiener integral [23]; that is, the integral is based on polygonal path approximations. In this section, we define different kinds of sequential function space integrals for functionals on the function space . In Section 4 below, we also adopt sequential approaches to define our function space transforms on . The sequential definition for the Feynman integral in [21] was defined as the limit of a sequence of finite dimensional Lebesgue integrals. Essentially, our sequential function space integrals and transforms are defined in terms of a sequence of complex measures on the function space .

Next, we introduce two sequential definitions for certain function space integrals on . Throughout the rest of this paper, let and denote the set of complex numbers with positive real part and nonzero complex numbers with nonnegative real part, respectively. Furthermore, for all , is always chosen to have nonnegative real part.

Let , , and be as in Section 2. For , and each , let

and let

Using (23) and (11), we observe that for all and every ,

We are now ready to state the definition of the sequential function space integrals.

Definition 6. Let be a measurable functional on . Let be a real number and let be a sequence of complex numbers in such that . If the following limit exists, one calls it the sequential -function space integral of with parameter , and we write

We also define the sequential -function space integral of with parameter by the formula

if it exists.

Let on and let

Then, since is a scale invariant null set, we have . But by the definition of the sequential function space integrals, we see that

Given two complex-valued measurable functionals and on , we will write if and furthermore if for all . The relation is clearly an equivalence relation.

Definition 7. Let be a functional on . If is -continuous for s-a.e. and every , then one says that is -s-continuous.

The functionals discussed in Examples 3 and 4 above are -s-continuous. Next we introduce a class of functionals which are -s-continuous.

Let be the space of complex-valued, countably additive (and hence finite) Borel measures on . is a Banach algebra under the total variation norm and with convolution as multiplication.

Definition 8. The Fresnel type class is the space of functionals on expressible in the form

for s-a.e. and every , where the associated measure is an element of . More precisely, since we will identify functionals which coincide under the relation on , can be regarded as the space of all ≊-equivalence classes of functionals of the form (36).

The Fresnel type class is a Banach algebra with norm

In fact, the correspondence is injective, carries convolution into pointwise multiplication, and is a Banach algebra isomorphism where and are related by (36).

Proposition 9. Let . Then is -s-continuous.

Proof. Let be given by (36), and for each , let be given by (19). Then substituting for , we have

By Example 4, the exponential in (38) approaches the exponential in equation (36) as . Note that for each , the exponential in (38) is measurable in on . Thus, by the bounded convergence theorem, we observe for s-a.e and every , and the proposition is proved.

The functional defined by equation (34) above is not -s-continuous. Thus, .

The following lemma plays a key role in the proofs of Propositions 12 and 13 below.

Lemma 10. For each , let be given by (19) and let and be given by (29) and (30), respectively. Then for all , all , and each , the function space integrals

exist and are given by (40) and (41) below, respectively.

Proof. Using (19), (29), (23), the Fubini theorem, and (11), we obtain

Also, using (19), (30), (23), the Fubini theorem, and (11), we obtain

Let be given by (36). Proceeding formally using (36), (40), and (41), we see that the sequential function space integrals and are given by the formulas

respectively.

For and , let

Then, for each , and are unbounded functions of . Hence, and might not exist. From this observation, we clearly need to impose additional restrictions, such as (45) below, on the functionals in for the existence of the sequential and -function space integrals of .

Let be a positive real number. We define a subclass of by if and only if

where and are related by (36).

The following example suggests the necessity of the condition (45) to ensure the existence of the sequential function space integrals of functionals in .

For each , let for . Consider a measure on which is concentrated on the set and for each . Then, is an element of and it follows that

Using the same method, we can find an example for the functional in that the sequential integrals do not exist.

Given a positive real number , let

Let be a real number with . Then, since

we can see that and are elements of the domain .

We also need the following lemma to obtain Propositions 12 and 13.

Lemma 11. Let and be as in Section 2, and let be a positive real number. Let be a sequence in such that , where is a real number with . Then there exists a sufficiently large such that for all ,

Proof. Let be a sequence in such that , and let be given by (47). Then, we observe that for each ,

Since , there exists a sufficiently large such that for every , and

Also, there exists a positive real number such that .

Let and be as in Section 2 above. Using Parseval’s identity, we observe

for . Also, using the Cauchy-Schwartz inequality, we have

Thus, there exists a sufficiently large such that for every ,

Using these facts, we obtain the inequality (49) and (50).

Proposition 12. Let be a positive real number and let be given by (36). Then for all real with , the sequential -function space integral exists and is given by the formula

where is given by (43) above.

Proof. Let be given by (36). First note that the equality in (36) holds for all . Let be a real number with , and let be a sequence in which converges to . Using (36), (19), (29), and the Fubini theorem, we obtain that for each ,

But, by (40) and (49), we know that the right hand side of (57) is dominated by (45) for all but a finite number of values of . Next, using (57), (40) with replaced with , the dominated convergence theorem, Parseval’s relation, and (43), we obtain

which concludes the proof of Proposition 12.

We establish our next proposition after careful examination of the proof of Proposition 12, and by using (30), (41), (44), and (50) instead of (29), (40), (43), and (49), respectively.

Proposition 13. Let and be as in Proposition 12. Then for all real with , the sequential -function space integral exists and is given by the formula

where is given by (44).

4. Sequential Function Space Transforms

In this section, we introduce two sequential transforms on the function space . We then establish that each of these transforms acts like an inverse transform of the other transform. Our definitions of the sequential transforms are based on the sequential function space integrals defined in Section 3 above.

Definition 14. Let be a nonzero real number. For , we define the sequential -function space transform of with parameter by the formula

if it exists. Also, we define the sequential -function space transform of with parameter by the formula

if it exists.

In Theorem 15 below, we establish the existence of the sequential -function space transform of functionals in .

Theorem 15. Let and be as in Proposition 12. Then for all real with , the sequential -function space transform of , exists and is an element of with associated measure defined by

where is given by (43), is the Borel -algebra, and is the associated measure of by (36). Furthermore, one sees that for s-a.e. and all ,

with

Proof. Let be given by (36), and for a real with , let be a sequence in which converges to . Proceeding as in the proof of Proposition 12, we obtain that for s-a.e. and all ,

Given with , let the set function be given by (62). Then, by the definition of the class , the inequality (64) holds. From this, we see that belongs to and that can be expressed as the third expression of (63) for s-a.e. and all . Thus, we conclude the proof of Theorem 15.

In Theorem 16 below, we also establish the existence of the sequential -function space transform of functionals in .

Theorem 16. Let and be as in Proposition 12. Then for all real with , the sequential -function space transform of , exists and is an element of with associated measure defined by

where is given by (44) and is the associated measure of by (36). Furthermore, one sees that for s-a.e. and all ,

with

In view of Theorems 15 and 16, we obtain the following assertion.

Theorem 17. Let be a positive real number and let be given by (36). Then

(i)for all real with , and are elements of , and one has

 and(ii)for all real with , and are elements of , and one has

We finish this section with some examples to apply our results. Let be the class of complex-valued, countably additive Borel measures on , the Borel class of . For , the Fourier transform of is a complex-valued function defined on by the formula

Let and let . Define by

Define a function by and let . It is quite clear that is in and is supported by , the subspace of spanned by . Now, for s-a.e. and all ,

Thus, is an element of .

Suppose that for a fixed positive real number ,

It is easy to show that condition (74) is equivalent to condition (45) above. Thus, applying Theorems 15 and 16, the sequential function space transforms and of exist for all with if the complex measure corresponding to satisfies condition (74).

Next, we present functionals in whose associated measures satisfy the condition (74).

Example 18. Let be the linear operator defined by . Then, the adjoint operator of is given by . Using an integration by parts formula, we have

Given real numbers and with , let be the Gaussian measure given by

Then and

The complex measure given by (76) satisfies condition (74) for all real . Thus, we can apply the results in Section 4 to the functional of the form

We note that and so that for every , the sequential transforms of , , and exist and are elements of by Theorems 15 and 16 above. Furthermore, we have that

for all real by Theorem 17. In fact, by a simple calculation, we obtain that for all ,

for s-a.e. and all .

In particular, if we choose , , and in (78), we have

Example 19. The functional

also is a functional under our consideration because

where is the Dirac measure concentrated at in . The Dirac measure also satisfies condition (45) for all real ; that is, .

5. Concluding Remarks

In this section, we examine certain aspects among the generalized analytic Fourier-Feynman transform [12, 13], the generalized integral transform [15, 16], and the sequential transforms for functionals on . As possible, we adopt the definitions and notation of [11–13, 15, 16] for the (generalized) analytic Fourier-Feynman transform and the (generalized) integral transform.

A major goal of the authors in [12, 13, 15, 16] was to generalize the concepts of the analytic Fourier-Feynman transform and the integral transform of the functionals of paths for the generalized Brownian motion process.

In [5, 6], the authors obtained the existence of the analytic Fourier-Feynman transform for several large classes of functionals on Wiener space . In particular, they showed that for all real ,

for s-a.e. . Thus, analytic Fourier-Feynman transform “” with parameter has the inverse transform “”. However, in view of the results in [12, 13], (84) does not hold for functionals on . As mentioned in Section 1, the existence of an inverse generalized Fourier-Feynman transform on has not yet been established. It is not easy to verify the existence of the inverse generalized Fourier-Feynman transform because the generalized Brownian motion process has a drift term .

In [9, 10], the authors studied the integral transform of functionals in . They showed that for and nonzero complex numbers and with , , , , and ,

where . That is to say, “” is given by “”. In [16], Chang et al. presented a version of inverse transform of the generalized integral transform as follows: for appropriate functionals on , the complexification of ,

for ; that is,

But they pointed out that for any nonzero complex numbers and , there are no nonzero complex numbers , and such that

Hence, the inverse generalized integral transform, , cannot be expressed as a simple integral transform.

Let us return to the singular integral transforms discussed in this paper. Theorem 17 tells us that the sequential function space transforms and have inverse transforms and , respectively. By choosing and on , the function space reduces to the Wiener space and the sequential transforms and coincide.

Let be as in Theorem 15. Then, by the definition of the generalized analytic Fourier-Feynman transform, we can see that of exists and

for s-a.e. and all . This fact tells us that the sequential -function space transform plays a prominent role as an inverse transform of the generalized analytic Fourier-Feynman transform .

Acknowledgments

The authors would like to express their gratitude to the referees for their valuable comments and suggestions which have improved the original paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2011-0014552) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2012R1A1A1004774). The authors declare that there is no conflict of interests regarding the publication of this paper.

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