# Sequential Generalized Transforms on Function Space

Authors: Jae Gil Choi; Hyun Soo Chung; Seung Jun Chang
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We define two sequential transforms on a function spaceCa,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].


---

## Body

## 1. Introduction and Preliminaries

LetC0[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 onCa,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 spaceCa,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.Leta(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 subsetB 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.LetLa,b2[0,T] 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,
(1)La,b2[0,T]={v:∫0T‍v2(s)db(s)<+∞,∫0T‍v2(s)d|a|(s)<+∞},
where |a|(·) is the total variation function of a(·). Then, (La,b2[0,T],∥·∥a,b) is a separable Hilbert space with inner product defined by
(2)(u,v)a,b=∫0T‍u(t)v(t)d[b(t)+|a|(t)].
Note that ∥u∥a,b=(u,u)a,b=0 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 La,b2[0,T]. If a(t)≡0 and b(t)=t, then La,b2[0,T]=L2[0,T]. In fact,
(3)(La,b2[0,T],∥·∥a,b)⊂(L0,b2[0,T],∥·∥0,b)=(L2[0,T],∥·∥2)
because the norms ∥·∥0,b and ∥·∥2 are equivalent.Let(4)Ca,b′[0,T]={∫0t‍w∈Ca,b[0,T]:w(t)=∫0t‍z(s)db(s)forsomez∈La,b2[0,T]}.
For w∈Ca,b′[0,T], with w(t)=∫0t‍z(s)db(s) for t∈[0,T], let D:Ca,b′[0,T]→La,b2[0,T] be defined by the formula
(5)Dw(t)=z(t)=w′(t)b′(t).
Then Ca,b′≡Ca,b′[0,T] with inner product
(6)(w1,w2)Ca,b′=∫0T‍Dw1(t)Dw2(t)db(t)=∫0T‍z1(t)z2(t)db(t)=(z1,z2)0,b
is a separable Hilbert space.Note that the two separable Hilbert spacesLa,b2[0,T] and Ca,b′[0,T] are homeomorphic under the linear operator given by (5). The inverse operator of D is given by
(7)(D-1z)(t)=∫0t‍z(s)db(s).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 ∫0T‍|a′(t)|2dt<+∞. 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
(8)∫0T‍|a′(t)|2db(t)=∫0T‍|a′(t)|2b′(t)dt<+∞.
But we cannot ensure that
(9)∫0T‍|a′(t)|2d|a|(t)<+∞
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 functiona:[0,T]→ℝ satisfies the requirement (9) if and only if a is an element of Ca,b′[0,T].Under the requirement (9), we observe that for each z∈La,b2[0,T],
(10)∫0T‍z(t)da(t)=∫0T‍Dw(t)Da(t)db(t)=(w,a)Ca,b′,
where w(t)=∫0t‍z(s)db(s) for t∈[0,T].The following integration formula is used several times in this paper:(11)∫ℝ‍exp{-αu2+βu}du=παexp{β24α}
for complex numbers α and β with Re(α)>0.


## 2. A Representation for Pathsx in Ca,b[0,T]

In this section, we investigate a representation for pathsx 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 Ca,b′[0,T], such that for each j=1,2,…, Dgj(t)=αj(t) is of bounded variation on [0,T]. Then, for each w∈Ca,b′[0,T] with Dw=z∈La,b2[0,T], we can write
(12)w(t)=∑j=1∞(w,gj)Ca,b′‍gj(t)=∑j=1∞‍(z,αj)0,bgj(t)
on [0,T].For eachw=D-1z∈Ca,b′[0,T], the PWZ type stochastic integral (w,x)Ca,b′~ is defined by the formula
(13)(w,x)Ca,b′~=limn→∞∫0T‍∑j=1n‍(w,gj)Ca,b′Dgj(t)dx(t)=limn→∞∫0T‍∑j=1n‍(z,αj)0,bαj(t)dx(t)
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 eachw∈Ca,b′[0,T], the PWZ type stochastic integral (w,x)Ca,b′~ exists for μ-a.e. x∈Ca,b[0,T].(2)
The PWZ type stochastic integral(w,x)Ca,b′~ is essentially independent of the choice of the complete orthonormal set {gj}j=1∞ in Ca,b′[0,T].(3)
It follows from the definition of the PWZ type stochastic integral and from Parseval’s equality that ifw∈Ca,b′[0,T] and x∈Ca,b′[0,T], then (w,x)Ca,b′~exists and we have (w,x)Ca,b′~=(w,x)Ca,b′.(4)
IfDw=z∈La,b[0,T] is of bounded variation on [0,T], then the PWZ type stochastic integral (w,x)Ca,b′~ equals the Riemann-Stieltjes integral ∫0T‍z(t)dx(t) 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 numberc, w∈Ca,b′[0,T] and x∈Ca,b[0,T], we have
(14)(w,cx)Ca,b′~=c(w,x)Ca,b′~=(cw,x)Ca,b′~.Thus, the assertions (1) and (4) hold for s-a.e. x∈Ca,b[0,T].(6)
For eachw∈Ca,b′[0,T], (w,x)Ca,b′~ is a Gaussian random variable with mean (w,a)Ca,b′ and variance ∥w∥Ca,b′2. For all w1,w2∈Ca,b′[0,T], we have
(15)∫Ca,b[0,T](w1,x)Ca,b′~(w2,x)Ca,b′~dμ(x)=(w1,w2)Ca,b′+(w1,a)Ca,b′(w2,a)Ca,b′.Now, we are ready to examine a representation for pathsx in Ca,b[0,T]. Throughout the rest of this paper, we will use the symbol 𝒢 for a complete orthonormal set {gj}j=1∞ in Ca,b′[0,T], such that for each j=1,2,…, Dgj=αj is of bounded variation on [0,T]. Then, for each j=1,2,…,
(16)γj(x)≡(gj,x)Ca,b′~=∫0T‍αj(t)dx(t)
is a Gaussian random variable with mean
(17)Aj≡(gj,a)Ca,b′=∫0T‍αj(t)da(t)
and variance
(18)∥gj∥Ca,b′2=∫0T‍αj2(t)db(t)=1.
We note that the set {γ1(x),γ2(x),…} forms a set of independent Gaussian random variables on Ca,b[0,T].Let𝒢≡{gj}j=1∞ be as above, and let x∈Ca,b[0,T]. For each n=1,2,…, define
(19)Zn(x)(t)≡∑j=1n‍γj(x)gj(t).
Then, Zn(x) is an element of Ca,b′[0,T] for all x∈Ca,b[0,T] and all n=1,2,….For eacht∈[0,T], let Ψt(s)=∫0s‍χ[0,t](τ)db(τ). Then, we observe that for each t∈[0,T],
(20)x(t)=∫0T‍χ[0,t](s)dx(s)=(Ψt,x)Ca,b′~=limn→∞∑j=1n‍(Ψt,gj)Ca,b′∫0T‍αj(s)dx(s)=limn→∞∑j=1n‍∫0t‍αj(s)db(s)γj(x)=limn→∞∑j=1n‍γj(x)gj(t)=limn→∞Zn(x)(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 Ca,b′[0,T]. If we choose the complete orthonormal sine sequence 𝒢≡{gj}j=1∞ in Ca,b′[0,T], where
(21)gj(t)=2b(T)(j-1/2)πsin[(j-1/2)πb(t)b(T)];
that is,
(22)αj(t)=Dgj(t)=2b(T)cos[(j-1/2)πb(t)b(T)]
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 spaceCa,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
(23)∫Ca,b[0,T]H(x)dμ(x)≡∫Ca,b[0,T]h(γ1(x),…,γn(x))dμ(x)=(2π)-n/2∫ℝnh(u1,…,un)×exp{-∑j=1n‍[uj-Aj]22}du1…dun
in the sense that if either side of (23) exists, both sides exist and equality holds.
LetF be a functional on Ca,b[0,T] and let 𝒢≡{gj}j=1∞ be a complete orthonormal set in Ca,b′[0,T]. Then, we say that F is 𝒢-continuous at x if
(24)limn→∞F(Zn(x))=F(x),
where Zn(x) is given by (19).Example 3.
For eacht∈[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 x∈Ca,b′[0,T].Example 4.
Givenw=D-1z∈Ca,b′[0,T], let F:Ca,b[0,T]→ℝ be given by F(x)=(w,x)Ca,b′~. Since (w,x)Ca,b′~=(w,x)Ca,b′ for every x∈Ca,b′[0,T], by the definition of PWZ type stochastic integral, we obtain that
(25)limn→∞F(Zn(x))=limn→∞(w,Zn(x))Ca,b′=limn→∞∫0T‍z(t)dZn(x)(t)=limn→∞∫0T‍z(t)d(∑j=1n‍γj(x)gj(t))=limn→∞∑j=1n‍γj(x)∫0T‍z(t)dgj(t)=limn→∞∑j=1n‍(z,αj)0,bγj(x)=(w,x)Ca,b′~=F(x)
for s-a.e. x∈Ca,b[0,T] and all x∈Ca,b′[0,T].Proposition 5.
LetF:Ca,b[0,T]→ℂ be μ-integrable. Assume that F is 𝒢-continuous for μ-a.e. x∈Ca,b[0,T]. Then
(26)∫Ca,b[0,T]‍F(x)dμ(x)=limn→∞∫Ca,b[0,T]‍F(Zn(x))dμ(x),
where Zn(x) is given by (19).Proof.
For eachn=1,2,…, let Fn(x)=F(Zn(x)). Then Fn is μ-integrable. By our assumption, we observe that
(27)limn→∞Fn(x)=limn→∞F(Zn(x))=F(x)
for μ-a.e. x∈Ca,b[0,T]. Thus, by the dominated convergence theorem, it follows that
(28)limn→∞∫Ca,b[0,T]‍F(Zn(x))dμ(x)=limn→∞∫Ca,b[0,T]‍Fn(x)dμ(x)=∫Ca,b[0,T]‍limn→∞Fn(x)dμ(x)=∫Ca,b[0,T]‍F(x)dμ(x)
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 onCa,b[0,T]. Throughout the rest of this paper, let ℂ+ and ℂ~+ 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.Let𝒢≡{gj}j=1∞, γj(x), and Aj be as in Section 2. For λ∈ℂ+, x∈Ca,b[0,T] and each n=1,2,…, let
(29)Wn+(λ;x)≡λn/2exp{(1-λ2)∑j=1n‍γj2(x)hhhhhhhhhhhhhhhh+(λ1/2-1)∑j=1n‍Ajγj(x)}
and let
(30)Wn-(λ;x)≡λn/2exp{(1-λ2)∑j=1n‍γj2(x)hhhhhhhhhhhhhhhh+((-λ)1/2-1)∑j=1n‍Ajγj(x)+∑j=1n‍Aj2}.Using (23) and (11), we observe that for all λ∈ℂ+ and every n=1,2,…,
(31)∫Ca,b[0,T]‍Wn+(λ;x)dμ(x)=∫Ca,b[0,T]‍Wn-(λ;x)dμ(x)=1.We are now ready to state the definition of the sequential function space integrals.Definition 6.
LetF be a measurable functional on Ca,b[0,T]. Let q≠0 be a real number and let {λn}n=1∞ 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
(32)∫Ca,b[0,T]Pq‍F(x)dsex=limn→∞∫Ca,b[0,T]‍F(Zn(x))Wn+(λn;x)dμ(x).
We also define the sequential𝒩-function space integral of F with parameter q by the formula
(33)∫Ca,b[0,T]Nq‍F(x)dsex=limn→∞∫Ca,b[0,T]‍F(Zn(x))Wn-(λn;x)dμ(x)
if it exists.
LetF1(x)≡1 on Ca,b[0,T] and let
(34)F2(x)={0,x∈Ca,b′[0,T],1,x∈Ca,b[0,T]-Ca,b′[0,T].
Then, since Ca,b′[0,T] is a scale invariant null set, we have F1≈F2. But by the definition of the sequential function space integrals, we see that
(35)∫Ca,b[0,T]Pq‍F1(x)dsex=∫Ca,b[0,T]Nq‍F1(x)dsex=1,∫Ca,b[0,T]Pq‍F2(x)dsex=∫Ca,b[0,T]Nq‍F2(x)dsex=0.
Given two complex-valued measurable functionalsF and G on Ca,b[0,T], we will write F≊G if F≈G and furthermore if F(x)=G(x) for all x∈Ca,b′[0,T]. The relation ≊ is clearly an equivalence relation.Definition 7.
LetF be a functional on Ca,b[0,T]. If F is 𝒢-continuous for s-a.e. x∈Ca,b[0,T] and every x∈Ca,b′[0,T], then one says that F is 𝒢-s-continuous.
The functionals discussed in Examples3 and 4 above are 𝒢-s-continuous. Next we introduce a class of functionals which are 𝒢-s-continuous.
Letℳ(Ca,b′[0,T]) be the space of complex-valued, countably additive (and hence finite) Borel measures on Ca,b′[0,T]. ℳ(Ca,b′[0,T]) is a Banach algebra under the total variation norm and with convolution as multiplication.Definition 8.
The Fresnel type classℱ*(Ca,b[0,T]) is the space of functionals on Ca,b[0,T] expressible in the form
(36)F(x)≊∫Ca,b′[0,T]‍exp{i(w,x)Ca,b′~}df(w)
for s-a.e. x∈Ca,b[0,T] and every x∈Ca,b′[0,T], where the associated measure f is an element of ℳ(Ca,b′[0,T]). 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 classℱ*(Ca,b[0,T]) is a Banach algebra with norm
(37)∥F∥=∥f∥=∫Ca,b′[0,T]‍d|f|(w).
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.
LetF∈ℱ*(Ca,b[0,T]). Then F is 𝒢-s-continuous.Proof.
LetF 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
(38)F(Zn(x))=∫Ca,b′[0,T]‍exp{i(w,Zn(x))Ca,b′~}df(w).
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 Ca,b′[0,T]×Ca,b[0,T]. Thus, by the bounded convergence theorem, we observe F(Zn(x))→F(x) for s-a.e x∈Ca,b[0,T] and every x∈Ca,b′[0,T], and the proposition is proved.The functionalF2 defined by equation (34) above is not 𝒢-s-continuous. Thus, F2∉ℱ*(Ca,b′[0,T]).The following lemma plays a key role in the proofs of Propositions12 and 13 below.Lemma 10.
For eachn=1,2,…, let Zn(x) be given by (19) and let Wn+ and Wn- be given by (29) and (30), respectively. Then for all w∈Ca,b′[0,T], all λ∈ℂ+, and each n=1,2,…, the function space integrals
(39)∫Ca,b[0,T]‍exp{i(w,Zn(x))Ca,b′~}Wn+(λ;x)dμ(x),∫Ca,b[0,T]‍exp{i(w,Zn(x))Ca,b′~}Wn-(λ;x)dμ(x)
exist and are given by (40) and (41) below, respectively.Proof.
Using (19), (29), (23), the Fubini theorem, and (11), we obtain
(40)∫Ca,b[0,T]‍exp{i(w,Zn(x))Ca,b′~}Wn+(λ;x)dμ(x)=(λ2π)n/2∫ℝn‍exp{-λ2∑j=1n‍uj2hhhhhhhhhhhhhhhh+∑j=1n‍(i(w,gj)Ca,b′+λ1/2Aj)ujhhhhhhhhhhhhhhhh-12∑j=1n‍Aj2}du1…dun=exp{-12λ∑j=1n‍(w,gj)Ca,b′2+iλ-1/2∑j=1n‍(w,gj)Ca,b′Aj}.
Also, using (19), (30), (23), the Fubini theorem, and (11), we obtain
(41)∫Ca,b[0,T]exp{i(w,Zn(x))C~a,b′}Wn-(λ;x)dμ(x)=exp{-12λ∑j=1n‍(w,gj)Ca,b′2-i(-λ)-1/2∑j=1n‍(w,gj)Ca,b′Aj}.LetF be given by (36). Proceeding formally using (36), (40), and (41), we see that the sequential function space integrals ∫Ca,b[0,T]Pq‍F(x)dsex and ∫Ca,b[0,T]Nq‍F(x)dsex are given by the formulas
(42)∫Ca,b[0,T]Pq‍F(x)dsex=∫Ca,b′[0,T]‍exp{i(-iq)-1/2-i2q∥w∥Ca,b′2hhhhhhhhhhhhh+i(-iq)-1/2(w,a)Ca,b′i2q}df(w),∫Ca,b[0,T]Nq‍F(x)dsex=∫Ca,b′[0,T]‍exp{-i2q∥w∥Ca,b′2hhhhhhhhhhhhh-i(iq)-1/2(w,a)Ca,b′i2q}df(w),
respectively.Forλ∈ℂ~+ and w∈Ca,b′[0,T], let
(43)ψP(λ;w)≡exp{-12λ∥w∥Ca,b′2+iλ-1/2(w,a)Ca,b′},(44)ψN(λ;w)≡exp{-12λ∥w∥Ca,b′2-i(-λ)-1/2(w,a)Ca,b′}.
Then, for each λ∈ℂ~+, |ψP(λ;w)| and |ψN(λ;w)| are unbounded functions of w. Hence, ∫Ca,b[0,T]Pq‍F(x)dsex and ∫Ca,b[0,T]Nq‍F(x)dsex 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.Letq0 be a positive real number. We define a subclass ℱq0*≡ℱq0*(Ca,b[0,T]) of ℱ*(Ca,b[0,T]) by F∈ℱq0* if and only if
(45)∫Ca,b′[0,T]‍exp{12q0∥w∥Ca,b′∥a∥Ca,b′}d|f|(w)<+∞,
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 eachn∈ℕ, let wn(t)=nb(t)=∫0t‍ndb(s) for t∈[0,T]. Consider a measure η on Ca,b′[0,T] which is concentrated on the set {wn:n∈ℕ} and η({wn})=1/(n2) for each n∈ℕ. Then, η is an element of ℳ(Ca,b′[0,T]) and it follows that
(46)∫Ca,b′[0,T]‍exp{12q0∥w∥Ca,b′∥a∥Ca,b′}d|η|(w)=∑n=1∞‍exp{12q0∥wn∥Ca,b′∥a∥Ca,b′}1n2=∑n=1∞‍exp{b(T)∥a∥Ca,b′2q0n}1n2≥∑n=1∞‍b(T)∥a∥Ca,b′2q01n=+∞.
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 numberq0, let
(47)Γq0={λ∈ℂ~+∣λ=α+iβ,|Im(λ-1/2)|<12q0}.
Let q be a real number with |q|>q0. Then, since
(48)(-iq)-1/2=12|q|+isign(q)2|q|,(iq)-1/2=12|q|-isign(q)2|q|,
we can see that -iq and iq are elements of the domain Int(Γq0).We also need the following lemma to obtain Propositions12 and 13.Lemma 11.
Let𝒢≡{gj}j=1∞ and Aj be as in Section 2, and let q0 be a positive real number. Let {λn}n=1∞ 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,
(49)exp{|Im(λn-1/2)||∑j=1n‍(gj,w)Ca,b′Aj|}<exp{12q0∥w∥Ca,b′∥a∥Ca,b′},(50)exp{|Im((-λn)-1/2)||∑j=1n‍(gj,w)Ca,b′Aj|}<exp{12q0∥w∥Ca,b′∥a∥Ca,b′}.Proof.
Let{λn}n=1∞={αn+iβn}n=1∞ be a sequence in ℂ+ such that λn→-iq, and let Γq0 be given by (47). Then, we observe that for each n∈ℕ,
(51)Im(λn-1/2)=-sign(βn)αn2+βn2-αn2(αn2+βn2),Im((-λn)-1/2)=sign(βn)αn2+βn2+αn2(αn2+βn2).
Since αn→0, there exists a sufficiently large K1∈ℕ such that for every n≥K1, λn∈Int(Γq0) and
(52)δ(q)=sup({12|q|}{|Im(λn-1/2)|:n≥K1}hhhhh∪{|Im((-λn)-1/2)|:n≥K1}∪{12|q|})<12q0.
Also, there exists a positive real number ε0>1 such that δ(q)<1/(ε02q0).
Let𝒢≡{gj}j=1∞ and Aj be as in Section 2 above. Using Parseval’s identity, we observe
(53)(w,a)Ca,b′=∑j=1∞‍(w,gj)Ca,b′(gj,a)Ca,b′=∑j=1∞‍(w,gj)Ca,b′Aj
for w∈Ca,b′[0,T]. Also, using the Cauchy-Schwartz inequality, we have
(54)-ε0∥w∥Ca,b′∥a∥Ca,b′<(w,a)Ca,b′<ε0∥w∥Ca,b′∥a∥Ca,b′.
Thus, there exists a sufficiently large K2∈ℕ such that for every n≥K2,
(55)|∑j=1n‍(gj,w)Ca,b′Aj|<ε0∥w∥Ca,b′∥a∥Ca,b′.
Using these facts, we obtain the inequality (49) and (50).Proposition 12.
Letq0 be a positive real number and let F∈ℱq0* be given by (36). Then for all real q with |q|>q0, the sequential 𝒫-function space integral ∫Ca,b[0,T]Pq‍F(x)dsex exists and is given by the formula
(56)∫Ca,b[0,T]Pq‍F(x)dsex=∫Ca,b′[0,T]‍ψP(-iq;w)df(w),
where ψP is given by (43) above.Proof.
LetF be given by (36). First note that the equality in (36) holds for all x∈Ca,b′[0,T]. Let q be a real number with |q|>q0, and let {λn}n=1∞ 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,…,
(57)∫Ca,b[0,T]‍F(Zn(x))Wn+(λn;x)dμ(x)=∫Ca,b′[0,T]‍∫Ca,b[0,T]‍exp{i(w,Zn(x))Ca,b′~}×Wn+(λn;x)dμ(x)df(w).
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
(58)limn→∞∫Ca,b[0,T]‍F(Zn(x))Wn+(λn;x)dμ(x)=∫Ca,b′[0,T]‍limn→∞exp{-12λn∑j=1n‍(w,gj)Ca,b′2hhhhhhhhhhhhhhhhh+iλn-1/2∑j=1n‍(w,gj)Ca,b′Aj}df(w)=∫Ca,b′[0,T]‍exp{-i2q∥w∥Ca,b′2hhhhhhhhhhhhh+i(-iq)-1/2(w,a)Ca,b′i2q}df(w),
which concludes the proof of Proposition 12.We establish our next proposition after careful examination of the proof of Proposition12, and by using (30), (41), (44), and (50) instead of (29), (40), (43), and (49), respectively.Proposition 13.
Letq0 and F∈ℱq0* be as in Proposition 12. Then for all real q with |q|>q0, the sequential 𝒩-function space integral ∫Ca,b[0,T]Nq‍F(x)dsex exists and is given by the formula
(59)∫Ca,b[0,T]Nq‍F(x)dsex=∫Ca,b′[0,T]‍ψN(-iq;w)df(w),
where ψN is given by (44).


## 4. Sequential Function Space Transforms

In this section, we introduce two sequential transforms on the function spaceCa,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.
Letq 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
(60)𝒫q(F)(y)=∫Ca,b[0,T]Pq‍F(y+x)dsex
if it exists. Also, we define the sequential 𝒩-function space transform 𝒩q(F) of F with parameter q by the formula
(61)𝒩q(F)(y)=∫Ca,b[0,T]Nq‍F(y+x)dsex
if it exists.
In Theorem15 below, we establish the existence of the sequential 𝒫-function space transform of functionals in ℱ*(Ca,b[0,T]).Theorem 15.
Letq0 and F∈ℱq0* 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 fqP defined by
(62)fqP(E)=∫E‍ψP(-iq;w)df(w),E∈ℬ(Ca,b′[0,T]),
where ψP is given by (43), ℬ(Ca,b′[0,T]) 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 y∈Ca,b′[0,T],
(63)𝒫q(F)(y)=∫Ca,b′[0,T]‍exp{i(w,y)Ca,b′~}ψP(-iq;w)df(w)=∫Ca,b′[0,T]‍exp{i(w,y)Ca,b′~}dfqP(w),
with
(64)∥fqP∥<∫Ca,b′[0,T]‍exp{12q0∥w∥Ca,b′∥a∥Ca,b′}d|f|(w)<+∞.Proof.
LetF be given by (36), and for a real q with |q|>q0, let {λn}n=1∞ 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 y∈Ca,b′[0,T],
(65)𝒫q(F)(y)=limn→∞∫Ca,b[0,T]‍F(y+Zn(x))Wn+(λn;x)dμ(x)=∫Ca,b′[0,T]‍exp{i(w,y)Ca,b′~}×limn→∞exp{-12λn∑j=1n‍(w,gj)Ca,b′2+iλn-1/2∑j=1n‍(w,gj)Ca,b′Aj}df(w)=∫Ca,b′[0,T]‍exp{i(w,y)Ca,b′~}ψP(-iq;w)df(w).
Givenq with |q|>q0, let the set function fqP:ℬ(Ca,b′[0,T])→ℂ be given by (62). Then, by the definition of the class ℱq0*, the inequality (64) holds. From this, we see that fqP belongs to ℳ(Ca,b′[0,T]) and that 𝒫q(F) can be expressed as the third expression of (63) for s-a.e. y∈Ca,b[0,T] and all y∈Ca,b′[0,T]. Thus, we conclude the proof of Theorem 15.In Theorem16 below, we also establish the existence of the sequential 𝒩-function space transform of functionals in ℱ*(Ca,b[0,T]).Theorem 16.
Letq0 and F∈ℱq0* 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 fqN defined by
(66)fqN(E)=∫E‍ψN(-iq;w)df(w),E∈ℬ(Ca,b′[0,T]),
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 y∈Ca,b′[0,T],
(67)𝒩q(F)(y)=∫Ca,b′[0,T]‍exp{i(w,y)Ca,b′~}ψN(-iq;w)df(w)=∫Ca,b′[0,T]‍exp{i(w,y)Ca,b′~}dfqN(w)
with
(68)∥fqN∥<∫Ca,b′[0,T]‍exp{12q0∥w∥Ca,b′∥a∥Ca,b′}d|f|(w)<+∞.In view of Theorems15 and 16, we obtain the following assertion.Theorem 17.
Letq0 be a positive real number and let F∈ℱq0/4* be given by (36). Then(i)
for all realq with |q|>q0, 𝒫q(F) and 𝒫-q(F) are elements of ℱq0*, and one has
(69)𝒩-q(𝒫q(F))≊F≊𝒩q(𝒫-q(F));and(ii)
for all realq with |q|>q0, 𝒩q(F) and 𝒩-q(F) are elements of ℱq0*, and one has
(70)𝒫-q(𝒩q(F))≊F≊𝒫q(𝒩-q(F)).We finish this section with some examples to apply our results. Letℳ(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 τ^ of τ is a complex-valued function defined on ℝ by the formula
(71)τ^(u)=∫ℝ‍exp{iuv}dτ(v).Leth∈Ca,b′[0,T] and let τ∈ℳ(ℝ). Define Fh,τ:Ca,b[0,T]→ℂ by
(72)Fh,τ(x)=τ^((h,x)Ca,b′~).
Define a function ϕ:ℝ→Ca,b′[0,T] by ϕ(v)=vh and let f=τ∘ϕ-1. It is quite clear that f is in ℳ(Ca,b′[0,T]) and is supported by [h], the subspace of Ca,b′[0,T] spanned by {h}. Now, for s-a.e. x∈Ca,b[0,T] and all x∈Ca,b′[0,T],
(73)Fh,τ(x)=∫ℝ‍exp{i(h,x)Ca,b′~v}dτ(v)=∫ℝ‍exp{i(ϕ(v),x)Ca,b′~}dτ(v)=∫Ca,b′[0,T]‍exp{i(w,x)Ca,b′~}d(τ∘ϕ-1)(w)=∫Ca,b′[0,T]‍exp{i(w,x)Ca,b′~}df(w).
Thus, Fh,τ is an element of ℱ*(Ca,b[0,T]).Suppose that for a fixed positive real numberq0>0,
(74)∫ℝ‍exp{(2q0)-1/2∥h∥Ca,b′∥a∥Ca,b′|v|}d|τ|(v)<+∞.
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 functionalsFh,τ in ℱ*(Ca,b[0,T]) whose associated measures satisfy the condition (74).Example 18.
LetS:Ca,b′[0,T]→Ca,b′[0,T] be the linear operator defined by Sw(t)=∫0t‍w(s)db(s). Then, the adjoint operator S* of S is given by S*w(t)=∫0t‍(w(T)-w(s))db(s). Using an integration by parts formula, we have
(75)(S*b,x)Ca,b′~=∫0T‍x(t)db(t).
Given real numbersm and σ2 with σ2>0, let τm,σ2 be the Gaussian measure given by
(76)τm,σ2(B)=(2πσ2)-1/2∫B‍exp{-(v-m)22σ2}dv,B∈ℬ(ℝ).
Then τm,σ2∈ℳ(ℝ) and
(77)τm,σ2^(u)=∫ℝexp{iuv}dτm,σ2(v)=exp{-12σ2u2+imu},u∈ℝ.
The complex measure τm,σ2 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
(78)Fh,τm,σ2(x)=τm,σ2^((h,x)Ca,b′~)=exp{-12σ2[(h,x)Ca,b′~]2+im(h,x)Ca,b′~}.
We note thatFh,τm,σ2∈∪q0>0ℱq0* and so that for every q∈ℝ-{0}, the sequential transforms of Fh,τm,σ2, 𝒫q(Fh,τm,σ2), and 𝒩q(Fh,τm,σ2) exist and are elements of ℱ(Ca,b[0,T]) by Theorems 15 and 16 above. Furthermore, we have that
(79)𝒩-q(𝒫q(Fh,τm,σ2))≊Fh,τm,σ2
for all real q∈ℝ-{0} by Theorem 17. In fact, by a simple calculation, we obtain that for all q∈ℝ-{0},
(80)𝒫q(Fh,τm,σ2)(y)=∫ℝ‍exp{i(h,y)Ca,b′~v-i2q∥h∥Ca,b′2v2+i(-iq)-1/2(h,a)Ca,b′vi2q}dτm,σ2(v),𝒩-q(Fh,τm,σ2)(y)=∫ℝ‍exp{i(h,y)Ca,b′~v+i2q∥h∥Ca,b′2v2-i(-iq)-1/2(h,a)Ca,b′v}dτm,σ2(v)
for s-a.e. y∈Ca,b[0,T] and all y∈Ca,b′[0,T].
In particular, if we chooseh=S*b, m=0, and σ2=2 in (78), we have
(81)FS*b,τ0,2(x)=exp{-(∫0T‍x(t)db(t))2}.Example 19.
The functional(82)FδS*b(x)=exp{i∫0T‍x(t)db(t)},x∈Ca,b[0,T]
also is a functional under our consideration because
(83)FδS*b(x)=exp{i(S*b,x)Ca,b′~}=∫Ca,b′[0,T]‍exp{i(w,x)Ca,b′~}dδS*b(w),
where δδS*b is the Dirac measure concentrated at S*b in Ca,b′[0,T]. The Dirac measure δδS*b also satisfies condition (45) for all real q0>0; that is, FδS*b∈∪q0>0ℱq0*.


## 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 Tq(2)(F) for several large classes of functionals F on Wiener space C0[0,T]. In particular, they showed that for all real q≠0,
(84)T-q(2)(Tq(2)(F))(y)=F(y)
for s-a.e. y∈C0[0,T]. Thus, L2 analytic Fourier-Feynman transform “Tq(2)” with parameter q has the inverse transform “T-q(2)”. 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, α=1-β2, and -π/4<arg(α)<π/4,
(85)ℱα′,1/βℱα,βF(y)=F(y),y∈C0[0,T],
where α′=1-1/β2. That is to say, “ℱα,β-1” 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],
(86)ℱ-iα,1ℱiα,1ℱ-α/β,1/βℱα,βF(y)=F(y)
for y∈Ka,b[0,T]; that is,
(87)ℱα,β-1=ℱ-iα,1ℱiα,1ℱ-α/β,1/β.But they pointed out that for any nonzero complex numbersα1,α2,β1 and β2, there are no nonzero complex numbers α′, and β′ such that
(88)ℱα1,β1ℱα2,β2=ℱα′,β′.
Hence, the inverse generalized integral transform, ℱα,β-1, cannot be expressed as a simple integral transform.Let us return to the singular integral transforms discussed in this paper. Theorem17 tells us that the sequential function space transforms 𝒫q and 𝒩q have inverse transforms 𝒫q-1=𝒩-q and 𝒩q-1=𝒫-q, 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.LetF be as in Theorem 15. Then, by the definition of the L2 generalized analytic Fourier-Feynman transform, we can see that Tq(2)(F) of F exists and
(89)Tq(2)(F)(y)=𝒫q(F)(y)
for s-a.e. y∈Ca,b[0,T] and all y∈Ca,b′[0,T]. 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 Tq(2).



---

*Source: 565832-2013-12-08.xml*