Abstract and Applied Analysis
Volume 2013, Issue 1 565832
Research Article
Open Access

Sequential Generalized Transforms on Function Space

Jae Gil Choi, 

Jae Gil Choi

Department of Mathematics, Dankook University, Cheonan 330-714, Republic of Korea dankook.ac.kr

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Hyun Soo Chung, 

Hyun Soo Chung

Department of Mathematics, Dankook University, Cheonan 330-714, Republic of Korea dankook.ac.kr

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Seung Jun Chang, 

Corresponding Author

Seung Jun Chang

Department of Mathematics, Dankook University, Cheonan 330-714, Republic of Korea dankook.ac.kr

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First published: 08 December 2013
https://doi.org/10.1155/2013/565832
Citations: 6
Academic Editor: Douglas Anderson

Abstract

We define two sequential transforms on a function space Ca,b[0, T] induced by generalized Brownian motion process. We then establish the existence of the sequential transforms for functionals in a Banach algebra of functionals on Ca,b[0, T]. 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 Ca,b[0, T].

1. Introduction and Preliminaries

Let C0[0, T] denote one-parameter Wiener space; that is, the space of all real-valued continuous functions x(t) on [0, T] with x(0) = 0. The study of the Fourier-Wiener transform of functionals on C0[0, T] 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 C0[0, T]. 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 Ca,b[0, T]. The function space Ca,b[0, T], 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 a(t). In case a(t) ≡ 0 and b(t) = t on [0, T], the general function space Ca,b[0, T] reduces to the Wiener space C0[0, T] 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 Ca,b[0, T] has not yet been established. It is a critical point that the generalized transforms on Ca,b[0, T] are essentially different from the transforms on Wiener space C0[0, T]. The main purpose of this paper is to define a transform on Ca,b[0, T] which has an inverse transform.

In this paper, we define two sequential transforms on the function space Ca,b[0, T]. To do this, we investigate a representation for sample paths of the generalized Brownian motion process and introduce the concept of the 𝒢-s-continuity for functionals on Ca,b[0, T]. We then proceed to establish the existence of the sequential transforms for functionals in a Banach algebra ℱ*(Ca,b[0, T]) of functionals on Ca,b[0, T]. 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 a(t) be an absolutely continuous real-valued function on [0, T] with a(0) = 0, a′(t) ∈ L2[0, T], and let b(t) be a strictly increasing, continuously differentiable real-valued function with b(0) = 0 and b′(t) > 0 for each t ∈ [0, T]. The generalized Brownian motion process Y determined by a(t) and b(t) is a Gaussian process with mean function a(t) and covariance function r(s, t) = min {b(s), b(t)}. For more details, see [12, 13]. By Theorem  14.2 in [18], the probability measure μ induced by Y, taking a separable version, is supported by Ca,b[0, T] (which is equivalent to the Banach space of continuous functions x on [0, T] with x(0) = 0 under the sup norm). Hence, (Ca,b[0, T], ℬ(Ca,b[0, T]), μ) is the function space induced by Y where ℬ(Ca,b[0, T]) is the Borel σ-algebra of Ca,b[0, T]. We then complete this function space to obtain (Ca,b[0, T], 𝒲(Ca,b[0, T]), μ), where 𝒲(Ca,b[0, T]) is the set of all Wiener measurable subsets of Ca,b[0, T].

A subset B of Ca,b[0, T] is said to be scale-invariant measurable provided ρB is 𝒲(Ca,b[0, T])-measurable for all ρ > 0, and a scale-invariant measurable set N is said to be a scale-invariant null set provided μ(ρN) = 0 for all ρ > 0. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s-a.e.). A functional F is said to be scale-invariant measurable provided F is defined on a scale-invariant measurable set, and F(ρ·) is 𝒲(Ca,b[0, T])-measurable for every ρ > 0. If two functionals F and G defined on Ca,b[0, T] are equal s-a.e., we write F ≈ G.

Let be the space of functions on [0, T] which are Lebesgue measurable and square integrable with respect to the Lebesgue-Stieltjes measures on [0, T] induced by a(·) and b(·); that is,
()
where |a | (·) is the total variation function of a(·). Then, is a separable Hilbert space with inner product defined by
()
Note that if and only if u(t) = 0 a.e. on [0, T]. Also note that all functions of bounded variation on [0, T] are elements of . If a(t) ≡ 0 and b(t) = t, then . In fact,
()
because the norms ∥·∥0,b and ∥·∥2 are equivalent.
Let
()
For , with for t ∈ [0, T], 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 D is given by
()
Recall that above, as well as in papers [12–16], we require that a : [0, T] → ℝ is an absolutely continuous function with a(0) = 0 and with . Our conditions on b : [0, T] → ℝ imply that δ < b′(t) < Δ for some positive real numbers δ and Δ, and b′(t) is continuous on [0, T]. Hence, we have
()
But we cannot ensure that
()
under current conditions. Note that the function a(t) = t2/3, 0 ≤ t ≤ T, does not satisfy condition (9) even though its derivative is an element of L2[0, T].

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

Lemma 1. The function a : [0, T] → ℝ satisfies the requirement (9) if and only if a is an element of .

Under the requirement (9), we observe that for each ,
()
where for t ∈ [0, T].
The following integration formula is used several times in this paper:
()
for complex numbers α and β with Re(α) > 0.

2. A Representation for Paths x in Ca,b[0, T]

In this section, we investigate a representation for paths x in Ca,b[0, T]. To do this, we first define a Paley-Wiener-Zygmund (PWZ) type stochastic integral.

Let {g1, g2, …} be a complete orthonormal set of functions in the separable Hilbert space , such that for each j = 1,2, …, Dgj(t) = αj(t) is of bounded variation on [0, T]. Then, for each with , we can write
()
on [0, T].
For each , the PWZ type stochastic integral is defined by the formula
()
for all x ∈ Ca,b[0, T] 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 μ-a.e. x ∈ Ca,b[0, T].

  • (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 Dw = z ∈ La,b[0, T] is of bounded variation on [0, T], then the PWZ type stochastic integral equals the Riemann-Stieltjes integral for μ-a.e. x ∈ Ca,b[0, T].

  • (5)

    The PWZ type stochastic integral has the expected linearity properties. That is, for any real number c, and x ∈ Ca,b[0, T], we have

    ()

  •  

    Thus, the assertions (1) and (4) hold for s-a.e. x ∈ Ca,b[0, T].

  • (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 x in Ca,b[0, T]. Throughout the rest of this paper, we will use the symbol 𝒢 for a complete orthonormal set in , such that for each j = 1,2, …, Dgj = αj is of bounded variation on [0, T]. Then, for each j = 1,2, …,
()
is a Gaussian random variable with mean
()
and variance
()
We note that the set {γ1(x), γ2(x), …} forms a set of independent Gaussian random variables on Ca,b[0, T].
Let be as above, and let x ∈ Ca,b[0, T]. For each n = 1,2, …, define
()
Then, Zn(x) is an element of for all x ∈ Ca,b[0, T] and all n = 1,2, ….
For each t ∈ [0, T], let . Then, we observe that for each t ∈ [0, T],
()
By the property of the PWZ type stochastic integral, the last expression of (20) converges for μ-a.e. x ∈ Ca,b[0, T].

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 𝒢 in . If we choose the complete orthonormal sine sequence in , where

()
that is,
()
for j = 1,2, …, then the corresponding Zn(x)(t) converges to x(t) uniformly in t with probability one. For more details, see [20] and the references therein.

We now state a fundamental integration formula on the function space Ca,b[0, T]. Let γj(x) and Aj be as above, let h : ℝn → ℂ be Lebesgue measurable, and let H(x) = h(γ1(x), …, γn(x)). Then

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

Let F be a functional on Ca,b[0, T] and let be a complete orthonormal set in . Then, we say that F is 𝒢-continuous at x if

()
where Zn(x) is given by (19).

Example 3. For each t ∈ [0, T], let F : Ca,b[0, T] → ℝ be given by F(x) = x(t). Then, using (20) and (14), we see that F is 𝒢-continuous for s-a.e. x ∈ Ca,b[0, T] and all .

Example 4. Given , let F : Ca,b[0, T] → ℝ be given by . Since for every , by the definition of PWZ type stochastic integral, we obtain that

()
for s-a.e. x ∈ Ca,b[0, T] and all .

Proposition 5. Let F : Ca,b[0, T] → ℂ be μ-integrable. Assume that F is 𝒢-continuous for μ-a.e. x ∈ Ca,b[0, T]. Then

()
where Zn(x) is given by (19).

Proof. For each n = 1,2, …, let Fn(x) = F(Zn(x)). Then Fn is μ-integrable. By our assumption, we observe that

()
for μ-a.e. x ∈ Ca,b[0, T]. 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 C0[0, T]. The sequential definition for the Feynman integral was intended to interpret the Feynman’s uniform measure [22] on path space C0[0, T], because there is no countably additive measure which weights all paths x in C0[0, T] equally in much the same way as Lebesgue measure weights all points in ℝ 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 Ca,b[0, T]. In Section 4 below, we also adopt sequential approaches to define our function space transforms on Ca,b[0, T]. 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 Ca,b[0, T].

Next, we introduce two sequential definitions for certain function space integrals on Ca,b[0, T]. Throughout the rest of this paper, let ℂ+ and denote the set of complex numbers with positive real part and nonzero complex numbers with nonnegative real part, respectively. Furthermore, for all λ ∈ ℂ, λ1/2 is always chosen to have nonnegative real part.

Let , γj(x), and Aj be as in Section 2. For λ ∈ ℂ+, x ∈ Ca,b[0, T] and each n = 1,2, …, let
()
and let
()
Using (23) and (11), we observe that for all λ ∈ ℂ+ and every n = 1,2, …,
()

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

Definition 6. Let F be a measurable functional on Ca,b[0, T]. Let q ≠ 0 be a real number and let be a sequence of complex numbers in ℂ+ such that λn → −iq. If the following limit exists, one calls it the sequential 𝒫-function space integral of F with parameter q, and we write

()

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

()
if it exists.

Let F1(x) ≡ 1 on Ca,b[0, T] and let

()
Then, since is a scale invariant null set, we have F1 ≈ F2. But by the definition of the sequential function space integrals, we see that
()

Given two complex-valued measurable functionals F and G on Ca,b[0, T], we will write F≊G if F ≈ G and furthermore if F(x) = G(x) for all . The relation ≊ is clearly an equivalence relation.

Definition 7. Let F be a functional on Ca,b[0, T]. If F is 𝒢-continuous for s-a.e. x ∈ Ca,b[0, T] and every , then one says that F is 𝒢-s-continuous.

The functionals discussed in Examples 3 and 4 above are 𝒢-s-continuous. Next we introduce a class of functionals which are 𝒢-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 ℱ*(Ca,b[0, T]) is the space of functionals on Ca,b[0, T] expressible in the form

()
for s-a.e. x ∈ Ca,b[0, T] and every , where the associated measure f is an element of . More precisely, since we will identify functionals which coincide under the relation ≊ on Ca,b[0, T], ℱ*(Ca,b[0, T]) can be regarded as the space of all ≊-equivalence classes of functionals of the form (36).

The Fresnel type class ℱ*(Ca,b[0, T]) is a Banach algebra with norm

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

Proposition 9. Let F ∈ ℱ*(Ca,b[0, T]). Then F is 𝒢-s-continuous.

Proof. Let F be given by (36), and for each n = 1,2, …, let Zn(x) be given by (19). Then substituting Zn(x) for x, we have

()
By Example 4, the exponential in (38) approaches the exponential in equation (36) as n → ∞. Note that for each n ∈ ℕ, the exponential in (38) is measurable in (w, x) on . Thus, by the bounded convergence theorem, we observe F(Zn(x)) → F(x) for s-a.e x ∈ Ca,b[0, T] and every , and the proposition is proved.

The functional F2 defined by equation (34) above is not 𝒢-s-continuous. Thus, .

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

Lemma 10. For each n = 1,2, …, let Zn(x) be given by (19) and let and be given by (29) and (30), respectively. Then for all , all λ ∈ ℂ+, and each n = 1,2, …, 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 F 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 , |ψP(λ; w)| and |ψN(λ; w)| are unbounded functions of w. Hence, and might not exist. From this observation, we clearly need to impose additional restrictions, such as (45) below, on the functionals F in ℱ*(Ca,b[0, T]) for the existence of the sequential 𝒫 and 𝒩-function space integrals of F.
Let q0 be a positive real number. We define a subclass of ℱ*(Ca,b[0, T]) by if and only if
()
where f and F 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 ℱ*(Ca,b[0, T]).

For each n ∈ ℕ, let for t ∈ [0, T]. Consider a measure η on which is concentrated on the set {wn : n ∈ ℕ} and η({wn}) = 1/(n2) for each n ∈ ℕ. Then, η is an element of and it follows that
()
Using the same method, we can find an example for the functional in ℱ*(Ca,b[0, T]) that the sequential integrals do not exist.
Given a positive real number q0, let
()
Let q be a real number with |q | > q0. Then, since
()
we can see that −iq and iq are elements of the domain .

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

Lemma 11. Let and Aj be as in Section 2, and let q0 be a positive real number. Let be a sequence in ℂ+ such that λ → −iq, where q is a real number with |q | > q0. Then there exists a sufficiently large K ∈ ℕ such that for all n ≥ K,

()
()

Proof. Let be a sequence in ℂ+ such that λn → −iq, and let be given by (47). Then, we observe that for each n ∈ ℕ,

()
Since αn → 0, there exists a sufficiently large K1 ∈ ℕ such that for every n ≥ K1, and
()
Also, there exists a positive real number ε0 > 1 such that .

Let and Aj 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 K2 ∈ ℕ such that for every n ≥ K2,
()

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

Proposition 12. Let q0 be a positive real number and let be given by (36). Then for all real q with |q | > q0, the sequential 𝒫-function space integral exists and is given by the formula

()
where ψP is given by (43) above.

Proof. Let F be given by (36). First note that the equality in (36) holds for all . Let q be a real number with |q| > q0, and let be a sequence in ℂ+ which converges to −iq. Using (36), (19), (29), and the Fubini theorem, we obtain that for each n = 1,2, …,

()
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 n. Next, using (57), (40) with λ replaced with λn, 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 q0 and be as in Proposition 12. Then for all real q with |q | > q0, the sequential 𝒩-function space integral exists and is given by the formula

()
where ψN is given by (44).

4. Sequential Function Space Transforms

In this section, we introduce two sequential transforms on the function space Ca,b[0, T]. 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 q be a nonzero real number. For y ∈ Ca,b[0, T], we define the sequential 𝒫-function space transform 𝒫q(F) of F with parameter q by the formula

()
if it exists. Also, we define the sequential 𝒩-function space transform 𝒩q(F) of F with parameter q by the formula
()
if it exists.

In Theorem 15 below, we establish the existence of the sequential 𝒫-function space transform of functionals in ℱ*(Ca,b[0, T]).

Theorem 15. Let q0 and be as in Proposition 12. Then for all real q with |q | > q0, the sequential 𝒫-function space transform of F, 𝒫q(F) exists and is an element of ℱ*(Ca,b[0, T]) with associated measure defined by

()
where ψP is given by (43), is the Borel σ-algebra, and f is the associated measure of F by (36). Furthermore, one sees that for s-a.e. y ∈ Ca,b[0, T] and all ,
()
with
()

Proof. Let F be given by (36), and for a real q with |q | > q0, let be a sequence in ℂ+ which converges to −iq. Proceeding as in the proof of Proposition 12, we obtain that for s-a.e. y ∈ Ca,b[0, T] and all ,

()

Given q with |q | > q0, 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 𝒫q(F) can be expressed as the third expression of (63) for s-a.e. y ∈ Ca,b[0, T] and all . Thus, we conclude the proof of Theorem 15.

In Theorem 16 below, we also establish the existence of the sequential 𝒩-function space transform of functionals in ℱ*(Ca,b[0, T]).

Theorem 16. Let q0 and be as in Proposition 12. Then for all real q with |q | > q0, the sequential 𝒩-function space transform of F, 𝒩q(F) exists and is an element of ℱ*(Ca,b[0, T]) with associated measure defined by

()
where ψN is given by (44) and f is the associated measure of F by (36). Furthermore, one sees that for s-a.e. y ∈ Ca,b[0, T] and all ,
()
with
()

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

Theorem 17. Let q0 be a positive real number and let be given by (36). Then

  • (i)

    for all real q with |q | > q0, 𝒫q(F) and 𝒫−q(F) are elements of , and one has

    ()

  •  

    and

  • (ii)

    for all real q with |q | > q0, 𝒩q(F) and 𝒩−q(F) are elements of , and one has

    ()

We finish this section with some examples to apply our results. Let ℳ(ℝ) be the class of complex-valued, countably additive Borel measures on ℬ(ℝ), the Borel class of ℝ. For τ ∈ ℳ(ℝ), the Fourier transform of τ is a complex-valued function defined on ℝ by the formula
()
Let and let τ ∈ ℳ(ℝ). Define Fh,τ : Ca,b[0, T] → ℂ by
()
Define a function by ϕ(v) = vh and let f = τ∘ϕ−1. It is quite clear that f is in and is supported by [h], the subspace of spanned by {h}. Now, for s-a.e. x ∈ Ca,b[0, T] and all ,
()
Thus, Fh,τ is an element of ℱ*(Ca,b[0, T]).
Suppose that for a fixed positive real number q0 > 0,
()
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 𝒫q(Fh,τ) and 𝒩q(Fh,τ) of Fh,τ exist for all q ∈ ℝ with |q | > q0 if the complex measure τ corresponding to Fh,τ satisfies condition (74).

Next, we present functionals Fh,τ in ℱ*(Ca,b[0, T]) whose associated measures satisfy the condition (74).

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

()

Given real numbers m and σ2 with σ2 > 0, let be the Gaussian measure given by

()
Then and
()
The complex measure given by (76) satisfies condition (74) for all real q0 > 0. Thus, we can apply the results in Section 4 to the functional of the form
()

We note that and so that for every q ∈ ℝ − {0}, the sequential transforms of , , and exist and are elements of ℱ(Ca,b[0, T]) by Theorems 15 and 16 above. Furthermore, we have that

()
for all real q ∈ ℝ − {0} by Theorem 17. In fact, by a simple calculation, we obtain that for all q ∈ ℝ − {0},
()
for s-a.e. y ∈ Ca,b[0, T] and all .

In particular, if we choose h = S*b, m = 0, and σ2 = 2 in (78), we have

()

Example 19. The functional

()
also is a functional under our consideration because
()
where is the Dirac measure concentrated at S*b in . The Dirac measure also satisfies condition (45) for all real q0 > 0; 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 Ca,b[0, T]. 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 L2 analytic Fourier-Feynman transform for several large classes of functionals F on Wiener space C0[0, T]. In particular, they showed that for all real q ≠ 0,
()
for s-a.e. y ∈ C0[0, T]. Thus, L2 analytic Fourier-Feynman transform “” with parameter q has the inverse transform “”. However, in view of the results in [12, 13], (84) does not hold for functionals F on Ca,b[0, T]. As mentioned in Section 1, the existence of an inverse generalized Fourier-Feynman transform on Ca,b[0, T] 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 a(t).
In [9, 10], the authors studied the integral transform of functionals F in L2(C0[0, T]). They showed that for F ∈ L2(C0[0, T]) and nonzero complex numbers α and β with |β | ≤ 1, β ≠ ±1, Re(1 − β2) > 0, , and −π/4 < arg(α) < π/4,
()
where . That is to say, “” is given by “ℱiα/β,1/β”. In [16], Chang et al. presented a version of inverse transform of the generalized integral transform ℱα,β as follows: for appropriate functionals F on Ka,b[0, T], the complexification of Ca,b[0, T],
()
for y ∈ Ka,b[0, T]; that is,
()
But they pointed out that for any nonzero complex numbers α1,   α2,   β1 and β2, there are no nonzero complex numbers α′, and β′ 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 𝒫q and 𝒩q have inverse transforms and , respectively. By choosing a(t) ≡ 0 and b(t) = t on [0, T], the function space Ca,b[0, T] reduces to the Wiener space C0[0, T] and the sequential transforms 𝒫q and 𝒩q coincide.

Let F be as in Theorem 15. Then, by the definition of the L2 generalized analytic Fourier-Feynman transform, we can see that of F exists and
()
for s-a.e. y ∈ Ca,b[0, T] and all . This fact tells us that the sequential 𝒩-function space transform 𝒩−q 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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