Relationship Between the Analytic Generalized Fourier–Feynman Transform and the Function Space Integral

Abstract

In this paper we investigated a relationship between the analytic generalized Fourier–Feynman transform associated with Gaussian process and the function space integral for exponential type functionals on the function space $C_{a,b}[0,T]$. The function space $C_{a,b}[0,T]$ can be induced by a generalized Brownian motion process. The Gaussian processes used in this paper are neither centered nor stationary.

1 Introduction

Let $C_0[0,T]$ denote the classical Wiener space. In [4], Cameron and Storvick defined the “sequential” Feynman integral by means of finite dimensional approximations for functionals on the Wiener space $C_0[0,T]$. The sequential definition for the Feynman path integral was intended to interpret the Feynman’s uniform measure [12] on continuous paths space $C_0[0,T]$, because there is no countably additive measure as Lebesgue measure. It is well known that there is generally no quasi-invariant measure on infinite-dimensional linear spaces, see [13]. Thus, the Cameron and Storvick’s sequential Feynman integral is a rigorous mathematical formulation for the Feynman’s path integral. On the other hand, the concept of the “analytic” Feynman integral on the Wiener space $C_0[0,T]$ was introduced by Cameron [1]. We refer to the reference [5, Section 1] for a heuristic structure of the analytic Feynman integral of functionals on $C_0[0,T]$. The analytic Feynman integral is not defined in terms of a countably additive nonnegative measure. Rather, they are defined in terms of a process of analytic continuation and a limiting procedure. In this reason, Cameron and Storvick provided the Banach algebra ${\mathcal {S}}$ of analytic Feynman integrable functionals in [2]. Since then, in [3], Cameron and Storvick expressed the analytic Feynman integral of functionals in ${\mathcal {S}}$ as the limit of a sequence of Wiener integrals.

Let $D=[0,T]$ and let $(\varOmega , {{\mathcal {B}}}, P)$ be a probability space. A generalized Brownian motion process (GBMP) on $\varOmega \times D$ is a Gaussian process $Y \equiv \{Y_t\}_{t\in D}$ such that $Y_0=c$ almost surely for some constant $c\in {\mathbb {R}}$ (in this paper we set $c=0$), and for any $0\le s< t\le T$,

$$\begin{aligned} Y_t -Y_s \sim N\big (a(t)-a(s), \,b(t)-b(s)\big ), \end{aligned}$$

where $N(m, \sigma ^2)$ denotes the normal distribution with mean m and variance $\sigma ^2$, a(t) is a continuous real-valued function on [0, T], and b(t) is a monotonically increasing continuous real-valued function on [0, T]. Thus, the GBMP Y is determined by the functions a(t) and b(t). For more details, see [14, 15]. Note that when $a(t)\equiv 0$ and $b(t)=t$, the GBMP is a standard Brownian motion (Wiener process). We are obliged to point out that a standard Brownian motion is stationary in time, whereas a GBMP is generally not stationary in time, and is subject to a drift a(t).

In [8, 10], the authors defined the analytic generalized Feynman integral and the analytic generalized Fourier–Feynman transform (GFFT) on the function space $C_{a,b}[0,T]$, and studied their properties and related topics. The function space $C_{a,b}[0,T]$, induced by a GBMP, was introduced by Yeh in [14], and was used extensively in [5,6,7,8,9,10,11].

In this paper we extend the ideas of [3] to the functionals on the very general function space $C_{a,b}[0,T]$. But our purpose of this paper is to obtain an expression of the analytic GFFT as a limit of a sequence of function space integrals on $C_{a,b}[0,T]$. The result in this paper enables us that the analytic GFFTs of functionals on the function space $C_{a,b}[0,T]$ can be interpreted as a limit of (non-analytic) function space transform.

The Wiener process used in [1,2,3,4] is centered and stationary in time and is free of drift. However, the Gaussian processes used in this paper, as well as in [6, 7], are neither centered nor stationary.

2 Preliminaries

In this section we first provide a brief background and some well-known results about the function space $C_{a,b}[0,T]$ induced by the GBMP.

Let a(t) be an absolutely continuous real-valued function on [0, T] with $a(0)=0$ and $a'(t)\in L^2[0,T]$, and let b(t) be an increasing and continuously differentiable real-valued function with $b(0)=0$ and $b'(t) >0$ for each $t \in [0,T]$. The GBMP 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 [5, 7, 8, 10, 14, 15]. By Theorem 14.2 in [15], the probability measure $\mu$ induced by Y, taking a separable version, is supported by $C_{a,b}[0,T]$ (which is equivalent to the Banach space of continuous functions x on [0, T] with $x(0)=0$ under the sup norm). Hence, $(C_{a,b}[0,T],{\mathcal {B}}(C_{a,b}[0,T]),\mu )$ is the function space induced by Y where ${\mathcal {B}}(C_{a,b}[0,T])$ is the Borel $\sigma$-field of $C_{a,b}[0,T]$. We then complete this function space to obtain the measure space $(C_{a,b}[0,T],{\mathcal {W}}(C_{a,b}[0,T]),\mu )$ where ${\mathcal {W}}(C_{a,b}[0,T])$ is the set of all $\mu$-Carathéodory measurable subsets of $C_{a,b}[0,T]$.

A subset B of $C_{a,b}[0,T]$ is said to be scale-invariant measurable provided $\rho B$ is ${\mathcal {W}}(C_{a,b}[0,T])$-measurable for all $\rho >0$, and a scale-invariant measurable set N is said to be scale-invariant null provided $\mu (\rho N)=0$ for all $\rho >0$. 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 $F(\rho \,\,\cdot \,)$ is ${\mathcal {W}}(C_{a,b}[0,T])$-measurable for every $\rho >0$. If two functionals F and G defined on $C_{a,b}[0,T]$ are equal s-a.e., we write $F\approx G$.

Remark 2.1

The function space $C_{a,b}[0,T]$ reduces to the Wiener space $C_0[0,T]$, considered in papers [1,2,3,4] if and only if $a(t)\equiv 0$ and $b(t) = t$ for all $t\in [0,T]$.

Let $L_{a,b}^2[0,T]$ (see [8] and [10]) 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 $a(\cdot )$ and $b(\cdot )$; i.e.,

$$\begin{aligned} L_{a,b}^2[0,T] =\bigg \{ v : \int _{0}^{T} v^2 (s) db(s)<+\infty \text{ and } \int _0^T v^2 (s) d |a|(s) < +\infty \bigg \} \end{aligned}$$

where $|a|(\cdot )$ denotes the total variation function of $a(\cdot )$. Then $L_{a,b}^2[0,T]$ is a separable Hilbert space with inner product defined by

$$\begin{aligned} (u,v)_{a,b}=\int _0^T u(t)v(t)dm_{|a|,b}(t)\equiv \int _0^T u(t)v(t)d[b(t)+|a|(t)], \end{aligned}$$

where $m_{|a|,b}$ denotes the Lebesgue–Stieltjes measure induced by $|a|(\cdot )$ and $b(\cdot )$. In particular, note that $\Vert u\Vert _{a,b}\equiv \sqrt{(u,u)_{a,b}} =0$ if and only if $u(t)=0$ a.e. on [0, T]. For more details, see [8, 10].

Let

$$\begin{aligned} C_{a,b}'[0,T] =\bigg \{ w \in C_{a,b}[0,T] : w(t)=\int _0^t z(s) d b(s) \hbox { for some } z \in L_{a,b}^2[0,T] \bigg \}. \end{aligned}$$

For $w\in C_{a,b}'[0,T]$, with $w(t)=\int _0^t z(s) d b(s)$ for $t\in [0,T]$, let $D: C_{a,b}'[0,T] \rightarrow L_{a,b}^2[0,T]$ be defined by the formula

$$\begin{aligned} Dw(t)= z(t)=\frac{w'(t)}{b'(t)}. \end{aligned}$$

(2.1)

Then $C_{a,b}' \equiv C_{a,b}'[0,T]$ with inner product

$$\begin{aligned} (w_1, w_2)_{C_{a,b}'} =\int _0^T Dw_1(t) Dw_2(t) d b(t) \end{aligned}$$

is a separable Hilbert space. For more details, see [6, 7].

Note that the two separable Hilbert spaces $L_{a,b}^2[0,T]$ and $C_{a,b}'[0,T]$ are (topologically) homeomorphic under the linear operator given by Eq. (2.1). The inverse operator of D is given by

$$\begin{aligned} (D^{-1}z)(t)=\int _0^t z(s) d b(s),\quad t\in [0,T]. \end{aligned}$$

In the case that $a(t)\equiv 0$, then the operator $D:C_{0,b}'[0,T]\rightarrow L_{0,b}^2[0,T]$ is an isometry.

In this paper, in addition to the conditions put on a(t) above, we now add the condition

$$\begin{aligned} \int _0^T |a'(t)|^2 d|a|(t)< +\infty \end{aligned}$$

(2.2)

from which it follows that

$$\begin{aligned} \begin{aligned} \int _0^T |Da(t)|^2d [b(t)+|a|(t)]&=\int _0^T \bigg |\frac{a'(t)}{b'(t)}\bigg |^2 d [b(t)+|a|(t)] \\&<M\Vert a'\Vert _{L^2[0,T]}+M^2\int _0^T |a'(t)|^2 d|a|(t) <+\infty , \end{aligned} \end{aligned}$$

where $M=\sup _{t\in [0,T]}(1/b'(t))$. Thus, the function $a: [0,T]\rightarrow {\mathbb {R}}$ satisfies the condition (2.2) if and only if $a(\cdot )$ is an element of $C_{a,b}'[0,T]$. Under the condition (2.2), we observe that for each $w\in C_{a,b}'[0,T]$ with $Dw=z$,

$$\begin{aligned} (w,a)_{C_{a,b}'}=\int _0^T Dw(t) Da(t) db(t)=\int _0^T z(t)da(t). \end{aligned}$$

Let $\{e_n\}_{n=1}^{\infty }$ be a complete orthonormal set in $(C_{a,b}'[0,T], \Vert \cdot \Vert _{C_{a,b}'})$ such that the $De_n$’s are of bounded variation on [0, T]. For $w\in C_{a,b}'[0,T]$ and $x\in C_{a,b}[0,T]$, we define the Paley–Wiener–Zygmund (PWZ) stochastic integral $(w,x)^{\sim }$ as follows:

$$\begin{aligned} (w,x)^{\sim } =\lim \limits _{n\rightarrow \infty }\int _0^T\sum _{j=1}^n(w,e_j)_{C_{a,b}'}De_j(t)dx(t) \end{aligned}$$

if the limit exists.

We will emphasize the following fundamental facts. For each $w\in C_{a,b}'[0,T]$, the PWZ stochastic integral $(w,x)^{\sim }$ exists for a.e. $x\in C_{a,b}[0,T]$. If $Dw=z\in L_{a,b}^2[0,T]$ is of bounded variation on [0, T], then the PWZ stochastic integral $(w,x)^{\sim }$ equals the Riemann–Stieltjes integral $\int _0^T z(t)dx(t)$. Furthermore, for each $w\in C_{a,b}'[0,T]$, $(w,x)^{\sim }$ is a Gaussian random variable with mean $(w,a)_{C_{a,b}'}$ and variance $\Vert w\Vert _{C_{a,b}'}^2$. Thus, for an orthogonal set $\{g_1,\ldots ,g_n\}$ of nonzero functions in $(C_{a,b}'[0,T],\Vert \cdot \Vert _{C_{a,b}'})$ and a Lebesgue measurable function $f:{\mathbb {R}}^n \rightarrow {\mathbb {C}}$, it follows that

$$\begin{aligned} \begin{aligned}&\int _{C_{a,b}[0,T]}f\big ((g_1,x)^{\sim },\ldots ,(g_n,x)^{\sim }\big )d\mu (x)\\&=\bigg (\prod _{j=1}^n2\pi \Vert g_j\Vert _{C_{a,b}'}^2\bigg )^{-n/2} \int _{{\mathbb {R}}^n} f(u_1,\ldots ,u_n)\\&\qquad \qquad \times \exp \bigg \{-\sum _{j=1}^{n}\frac{ [u_j-(g_j,a)_{C_{a,b}'}]^2 }{2\Vert g_j\Vert _{C_{a,b}'}^2}\bigg \} du_1\cdots du_n \end{aligned} \end{aligned}$$

(2.3)

in the sense that if either side of Eq. (2.3) exists, both sides exist and equality holds. Also we note that for $w, x\in C_{a,b}'[0,T]$, $(w,x)^{\sim }=(w,x)_{C_{a,b}'}$.

The following integration formula on the function space $C_{a,b}[0,T]$ is also used in this paper:

$$\begin{aligned} \int _{\mathbb {R}} \exp \big \{-\alpha u^2+ \beta u\big \}du =\sqrt{\frac{\pi }{\alpha }}\exp \Big \{\frac{\beta ^2}{4\alpha }\Big \} \end{aligned}$$

(2.4)

for complex numbers $\alpha$ and $\beta$ with $\mathrm{Re}(\alpha )> 0$.

3 Gaussian Processes

Let $C_{a,b}^*[0,T]$ be the set of functions k in $C_{a,b}'[0,T]$ such that Dk is continuous except for a finite number of finite jump discontinuities and is of bounded variation on [0, T]. For any $w\in C_{a,b}'[0,T]$ and $k\in C_{a,b}^*[0,T]$, let the operation $\odot$ between $C_{a,b}'[0,T]$ and $C_{a,b}^*[0,T]$ be defined by

$$\begin{aligned} w\odot k =D^{-1}(DwDk), \end{aligned}$$

where DwDk denotes the pointwise multiplication of the functions Dw and Dk. Then $(C_{a,b}^*[0,T],\odot )$ is a commutative algebra with the identity b. For a more detailed study of the operation $\odot$, see [7].

For each $t\in [0,T]$, let $\varPhi _t(\tau )=D^{-1}\chi _{[0,t]}(\tau )=\int _0^{\tau }\chi _{[0,t]}(u)db(u)$, $\tau \in [0,T]$, and for $k\in C_{a,b}'[0,T]$ with $Dk\ne 0$ $m_L$-a.e. on [0, T] ($m_L$ denotes the Lebesgue measure on [0, T]), let ${\mathcal {Z}}_{k}(x,t)$ be the PWZ stochastic integral

$$\begin{aligned} {\mathcal {Z}}_{k}(x,t)=(k\odot \varPhi _t ,x)^{\sim }, \end{aligned}$$

(3.1)

let $\beta _k(t)=\int _0^t \{Dk (u)\}^2db(u)$, and let $\alpha _k(t)=\int _0^t Dk (u)da(u)$. Then ${\mathcal {Z}}_k:C_{a,b}[0,T]\times [0,T]\rightarrow {\mathbb {R}}$ is a Gaussian process with mean function

$$\begin{aligned} \int _{C_{a,b}[0,T]}{\mathcal {Z}}_k(x,t)d\mu (x) =\int _0^t Dk (u) da(u) =\alpha _k(t) \end{aligned}$$

and covariance function

$$\begin{aligned} \begin{aligned}&\int _{C_{a,b}[0,T]}\big ({\mathcal {Z}}_k(x,s)-\alpha _k(s)\big ) \big ({\mathcal {Z}}_k(x,t)-\alpha _k(t)\big )d\mu (x)\\&=\int _0^{\min \{s,t\}} \{Dk (u)\}^2db(u)=\beta _k(\min \{s,t\}). \end{aligned} \end{aligned}$$

In addition, by [15, Theorem 21.1], ${\mathcal {Z}}_{k}(\cdot ,t)$ is stochastically continuous in t on [0, T]. If Dk is of bounded variation on [0, T], then, for all $x\in C_{a,b}[0,T]$, ${\mathcal {Z}}_{k}(x,t)$ is continuous in t. Also, for any functions $k_1$ and $k_2$ in $C_{a,b}'[0,T]$,

$$\begin{aligned} \begin{aligned}&\int _{C_{a,b}[0,T]}{\mathcal {Z}}_{k_1}(x,s){\mathcal {Z}}_{k_2}(x,t)d \mu (x)\\&=\int _0^{\min \{s,t\}}Dk_1(u)Dk_2(u)db(u) +\int _0^{s}Dk_1(u)da(u)\int _0^{t}Dk_2(u)da(u). \end{aligned} \end{aligned}$$

Of course if $k(t)\equiv b(t)$, then ${\mathcal {Z}}_b(x,t)=x(t)$, the continuous sample paths of the GBMP Y, of which the function space $C_{a,b}[0,T]$ consists. Choosing $a(t)\equiv 0$ and $b(t)=t$ on [0, T], as commented in Remark 2.1 above, the function space $C_{a,b}[0,T]$ reduces to the classical Wiener space $C_0[0,T]$, and thus the Gaussian process (3.1) with $k(t)\equiv t$ is an ordinary Wiener process.

From the properties of the PWZ stochastic integral and the operation $\odot$ between $C_{a,b}'[0,T]$ and $C_{a,b}^*[0,T]$, it follows that for all $\rho \in {\mathbb {R}}$,

$$\begin{aligned} \rho {\mathcal {Z}}_k(x,t) ={\mathcal {Z}}_{\rho k}(x,t) ={\mathcal {Z}}_{k}(\rho x,t), \end{aligned}$$

and for any $w\in C_{a,b}'[0,T]$ and each $k\in C_{a,b}^*[0,T]$,

$$\begin{aligned} (w,{\mathcal {Z}}_k(x,\cdot ))^{\sim } =(w\odot k,x)^{\sim } \end{aligned}$$

(3.2)

for $\mu$-a.e. $x\in C_{a,b}[0,T]$. Thus, throughout the remainder of this paper, we require k to be in $C_{a,b}^*[0,T]$ for each process ${\mathcal {Z}}_k$.

We define a class of those functions as follows: let

$$\begin{aligned} \mathrm {Supp}_{C_{a,b}^*}[0,T]=\{k\in C_{a,b}^*[0,T] : Dk\ne 0\,\, m_L\text{-a.e } \text{ on } [0,T]\}. \end{aligned}$$

Then for any $k\in \mathrm {Supp}_{C_{a,b}^*}[0,T]$, the Lebesgue–Stieltjes integrals

$$\begin{aligned} \Vert w\odot k\Vert _{C_{a,b}'}^2 =\int _0^T (Dw(t))^2(Dk(t))^2db(t) \end{aligned}$$

and

$$\begin{aligned} (w\odot k,a)_{C_{a,b}'} =\int _0^T Dw(t) Dk(t) Da(t)db(t)=\int _0^T Dw(t) Dk(t)da(t) \end{aligned}$$

exist for all $w\in C_{a,b}'[0,T]$.

4 Transforms on the Class of Exponential-Type Functionals

Given a function $k\in \mathrm {Supp}_{C_{a,b}^*}[0,T]$, we define the generalized ${\mathcal {Z}}_k$-function space integral (namely, the function space integral associated with the Gaussian paths ${\mathcal {Z}}_k(x,\cdot )$) for functionals F on $C_{a,b}[0,T]$ by the formula

$$\begin{aligned} I_k[F]\equiv I_{k,x}[F({\mathcal {Z}}_k (x,\cdot ))] =\int _{C_{a,b}[0,T]}F\big ({\mathcal {Z}}_k(x,\cdot )\big ) d \mu (x). \end{aligned}$$

Throughout this paper, let ${\mathbb {C}}$, ${\mathbb {C}}_+$ and $\mathbb {{\widetilde{C}}}_+$ denote the set of complex numbers, complex numbers with positive real part, and non-zero complex numbers with nonnegative real part, respectively. Furthermore, for each $\lambda \in {\mathbb {C}}$, $\lambda ^{1/2}$ denotes the principal square root of $\lambda$, i.e., $\lambda ^{1/2}$ is always chosen to have nonnegative real part.

Definition 4.1

Given a function $k\in \mathrm {Supp}_{C_{a,b}^*}[0,T]$, let ${\mathcal {Z}}_k$ be the Gaussian process given by (3.1) and let F be a ${\mathbb {C}}$-valued scale-invariant measurable functional on $C_{a,b}[0,T]$ such that

$$\begin{aligned} J_F({\mathcal {Z}}_k;\lambda )=I_{k,x} [F (\lambda ^{-1/2}{\mathcal {Z}}_k(x,\cdot ))] \end{aligned}$$

exists and is finite for all $\lambda >0$. If there exists a function $J_F^*({\mathcal {Z}}_k;\lambda )$ analytic on ${\mathbb {C}}_+$ such that $J_F^*({\mathcal {Z}}_k;\lambda )=J_F({\mathcal {Z}}_k;\lambda )$ for all $\lambda \in (0,+\infty )$, then $J_F^* ({\mathcal {Z}}_k;\lambda )$ is defined to be the analytic ${\mathcal {Z}}_k$-function space integral (namely, the analytic function space integral associated with the Gaussian paths ${\mathcal {Z}}_k(x,\cdot )$) of F over $C_{a,b}[0,T]$ with parameter $\lambda$, and for $\lambda \in {\mathbb {C}}_+$ we write

$$\begin{aligned} I_{k}^{\mathrm{an}_{\lambda }}[F] \equiv I_{k,x}^{\mathrm{an}_{\lambda }}[F({\mathcal {Z}}_k(x,\cdot ))] \equiv \int _{C_{a,b}[0,T]}^{\mathrm{an}_{\lambda }} F\big ({\mathcal {Z}}_k(x,\cdot )\big )d\mu (x): = J_F^*({\mathcal {Z}}_k;\lambda ). \end{aligned}$$

(4.1)

Let q be a non-zero real number and let F be a measurable functional whose analytic ${\mathcal {Z}}_k$-function space integral $J_F^*({\mathcal {Z}}_k;\lambda )$ exists for all $\lambda$ in ${\mathbb {C}}_+$. If the following limit exists, we call it the analytic generalized ${\mathcal {Z}}_k$-Feynman integral (namely, the analytic generalized Feynman integral associated with the paths ${\mathcal {Z}}_k(x,\cdot )$) of F with parameter q and we write

$$\begin{aligned} I_{k}^{\mathrm{anf}_{q}}[F] \equiv I_{k,x}^{\mathrm{anf}_{q}}[F({\mathcal {Z}}_k(x,\cdot ))] =\lim _{\lambda \rightarrow -iq} I_{k,x}^{\mathrm{an}_{\lambda }}[F({\mathcal {Z}}_k(x,\cdot ))], \end{aligned}$$

(4.2)

where $\lambda$ approaches $-iq$ through values in ${\mathbb {C}}_+$.

Next we state the definition of the analytic GFFT associated with Gaussian process on function space.

Definition 4.2

Given a function $k\in \mathrm {Supp}_{C_{a,b}^*}[0,T]$, let ${\mathcal {Z}}_k$ be the Gaussian process given by (3.1) and let F be a scale-invariant measurable functional on $C_{a,b}[0,T]$ such that for all $\lambda \in {\mathbb {C}}_+$ and $y \in C_{a,b}[0,T]$, the following analytic ${\mathcal {Z}}_k$-function space integral

$$\begin{aligned} T_{\lambda ,k}(F)(y) =I_{k,x}^{\mathrm { an}_{\lambda }}[F(y+{\mathcal {Z}}_k(x,\cdot ))] \end{aligned}$$

exists. Let q be a non-zero real number. For $p\in (1,2]$, we define the $L_p$ analytic ${\mathcal {Z}}_k$-GFFT (namely, the GFFT associated with the paths ${\mathcal {Z}}_k(x,\cdot )$), $T^{(p)}_{q,k}(F)$ of F, by the formula,

$$\begin{aligned} T^{(p)}_{q,k}(F)(y) = \mathop {{\text {l.i.m.}}}\limits _{\begin{array}{c} \lambda \rightarrow -iq \\ \,\, \lambda \in {\mathbb {C}}_+ \end{array}} T_{\lambda ,k} (F)(y) \end{aligned}$$

if it exists; i.e., for each $\rho >0$,

$$\begin{aligned} \mathop {\lim }\limits _{\begin{array}{c} \lambda \rightarrow -iq \\ \,\,\lambda \in {\mathbb {C}}_+ \end{array}} \int _{C_{a,b}[0,T]} \big | T_{\lambda ,k} (F)(\rho y)-T^{(p)}_{q,k}(F)(\rho y) \big |^{p'} d\mu (y)=0 \end{aligned}$$

where $1/p +1/p' =1$. We define the $L_1$ analytic ${\mathcal {Z}}_k$-GFFT, $T_{q,k}^{(1)}(F)$ of F, by the formula

$$\begin{aligned} T_{q,k}^{(1)}(F)(y) = \mathop {\lim }\limits _{\begin{array}{c} \lambda \rightarrow -iq \\ \,\, \lambda \in {\mathbb {C}}_+ \end{array}} T_{\lambda ,k}(F)(y) =I_{k,x}^{\mathrm { anf}_{q}}[F(y+{\mathcal {Z}}_k(x,\cdot ))] \end{aligned}$$

(4.3)

for s-a.e. $y\in C_{a,b}[0,T]$, if it exists.

We note that for $1\le p \le 2$, $T_{q,k}^{(p)}(F)$ is defined only s-a.e.. We also note that if $T_{q,k}^{(p)}(F)$ exists and if $F\approx G$, then $T_{q,k}^{(p)}(G)$ exists and $T_{q,k}^{(p)}(G)\approx T_{q,k}^{(p)}(F)$. Moreover, from Eqs. (4.2), (4.1), and (4.3), it follows that

$$\begin{aligned} I_{k}^{\mathrm { anf}_{q}}[F] \equiv I_{k,x}^{\mathrm { anf}_{q}}[F( {\mathcal {Z}}_k (x,\cdot ))] =T_{q,k}^{(1)}(F)(0) \end{aligned}$$

(4.4)

in the sense that if either side exists, then both sides exist and equality holds.

Remark 4.3

Note that if $k\equiv b$ on [0, T], then the analytic generalized ${\mathcal {Z}}_b$-Feynman integral, $I_{b}^{\mathrm { anf}_q}[F]$, and the $L_p$ analytic ${\mathcal {Z}}_b$-GFFT, $T_{q, b}^{(p)}(F)$ agree with the previous definitions of the analytic generalized Feynman integral and the analytic GFFT respectively [5, 8, 10].

Let ${\mathcal {E}}$ be the class of all functionals which have the form

$$\begin{aligned} \varPsi _w (x) =\exp \{(w,x)^{\sim } \} \end{aligned}$$

(4.5)

for some $w\in C_{a,b}'[0,T]$ and for s-a.e. $x\in C_{a,b}[0,T]$. More precisely, since we shall identify functionals which coincide s-a.e. on $C_{a,b}[0,T]$, the class ${\mathcal {E}}$ can be regarded as the space of all s-equivalence classes of functionals of the form (4.5).

Given $q\in {\mathbb {R}}\setminus \{0\}$, $\tau \in C_{a,b}'[0,T]$, and $k\in C_{a,b}^*[0,T]$, let ${\mathcal {E}}_{q,\tau ,k}$ be the class of all functionals having the form

$$\begin{aligned} \varPsi _{w}^{q,\tau ,k} (x) =K^a_{q,\tau , k}\varPsi _w (x) \end{aligned}$$

(4.6)

for s-a.e. $x\in C_{a,b}[0,T]$, where $\varPsi _w$ is given by Eq. (4.5) and $K^a_{q,\tau , k}$ is a complex number given by

$$\begin{aligned} K^a_{q,\tau , k}= \exp \bigg \{\frac{i}{2q}\Vert \tau \odot k\Vert _{C_{a,b}'}^2 +(-iq)^{-1/2}(\tau \odot k,a)_{C_{a,b}'} \bigg \}. \end{aligned}$$

(4.7)

The functionals given by Eq. (4.6) and linear combinations (with complex coefficients) of the $\varPsi _{w}^{q,\tau ,k}$’s are called the (partially) exponential-type functionals on $C_{a,b}[0,T]$.

For notational convenience, let $\varPsi _{w}^{0,\tau ,k} (x)=\varPsi _{w} (x)$ and let ${\mathcal {E}}_{0,\tau ,k}={\mathcal {E}}$. Then for any $(q,\tau ,k)\in {\mathbb {R}}\times C_{a,b}'[0,T]\times C_{a,b}^*[0,T]$, the class ${\mathcal {E}}_{q,\tau ,k}$ is dense in $L_2(C_{a,b}[0,T])$, see [9, 11]. We then define the class ${\mathcal {E}}(C_{a,b}[0,T])$ to be the linear span of ${\mathcal {E}}$, i.e., ${\mathcal {E}}(C_{a,b}[0,T])=\text {Span}{\mathcal {E}}$.

Remark 4.4

• (i) One can see that ${\mathcal {E}}(C_{a,b}[0,T])=\text {Span}{\mathcal {E}}_{q,\tau ,k}$ for every $(q,\tau ,k)\in {\mathbb {R}}\times C_{a,b}'[0,T]\times C_{a,b}^*[0,T]$.

• (ii) The linear space ${\mathcal {E}} (C_{a,b}[0,T])$ is a commutative (complex) algebra under the pointwise multiplication and with identity $\varPsi _0 \equiv 1$ because

  $$\begin{aligned} \varPsi _{w_1}^{q_1,\tau _1,k_1} (x) \varPsi _{w_2}^{q_2,\tau _2,k_2} (x) =K^a_{q_1,\tau _1, k_1}K^a_{q_2,\tau _2, k_2}\varPsi _{w_1+w_2} (x) \end{aligned}$$

  for $\mu$-a.e. $x\in C_{a,b}[0,T]$.

• (iii) Note that every exponential-type functional is scale-invariant measurable. Since we shall identify functionals which coincide s-a.e. on $C_{a,b}[0,T]$, ${\mathcal {E}} (C_{a,b}[0,T])$ can be regarded as the space of all s-equivalence classes of exponential-type functionals.

The following two theorems are due to by Chang and Choi [6].

Theorem 4.5

Let $\varPsi _w\in {\mathcal {E}}$ be given by Eq. (4.5). Then for all $p\in [1,2]$, any non-zero real number q, and each function k in $\mathrm {Supp}_{C_{a,b}^*}[0,T]$, the $L_p$ analytic ${\mathcal {Z}}_{k}$-GFFT of $\varPsi _w$, $T^{(p)}_{q,k}(\varPsi _w)$ exists and is given by the formula

$$\begin{aligned} T^{(p)}_{q, k}(\varPsi _w) \approx \varPsi _{w}^{q,w,k}, \end{aligned}$$

(4.8)

where $\varPsi _{w}^{q,w,k}$ is given by Eq. (4.6) with $\tau$ replaced with w. Thus, $T^{(p)}_{q,k}\!(\varPsi _w\!)$ is an element of ${\mathcal {E}}(\!C_{a,b}[0,T]\!)$.

Let F be a functional in ${\mathcal {E}}(C_{a,b}[0,T])$. Since ${\mathcal {E}}(C_{a,b}[0,T])=\text {Span}{\mathcal {E}}$, there exist a finite sequence $\{w_1,\ldots , w_n\}$ of functions in $C_{a,b}'[0,T]$, and a sequence $\{c_1,\ldots , c_n\}$ in ${\mathbb {C}}\setminus \{0\}$ such that

$$\begin{aligned} F \approx \sum _{j=1}^n c_j \varPsi _{w_j}. \end{aligned}$$

(4.9)

Then for all $p\in [1,2]$, any non-zero real number q, and each function k in $\mathrm {Supp}_{C_{a,b}^*}[0,T]$, the $L_p$ analytic ${\mathcal {Z}}_{k}$-GFFT of F, $T^{(p)}_{q,k}(F)$ exists and is given by the formula

$$\begin{aligned} \begin{aligned} T^{(p)}_{q,k}(F) \approx \sum _{j=1}^n c_j T^{(p)}_{q,k}(\varPsi _{w_j}) \approx \sum _{j=1}^n c_j \varPsi _{w_j}^{q,w_j,k}, \end{aligned} \end{aligned}$$

where $\varPsi _{w_j}^{q,w_j,k}$ is given by Eq. (4.6) with $\tau$ and w replaced with $w_j$ and $w_j$, for each $j\in \{1,\ldots ,n\}$, respectively.

Theorem 4.6

For all $p\in [1,2]$, any $q\in {\mathbb {R}}\setminus \{0\}$, and each $k\in \mathrm {Supp}_{C_{a,b}^*}[0,T]$, the $L_p$ analytic ${\mathcal {Z}}_{k}$-GFFT, $T_{q,k}^{(p)}: {\mathcal {E}}(C_{a,b}[0,T]) \rightarrow {\mathcal {E}}(C_{a,b}[0,T])$ is an onto transform.

5 Relationship Between the ${\mathcal {Z}}_k$-Fourier–Feynman Transform and the Function Space Integral

In this section, we establish a relationship between the analytic ${\mathcal {Z}}_k$-GFFT and the ${\mathcal {Z}}_k$-function space integral of functionals in the class ${\mathcal {E}}(C_{a,b}[0,T])$.

Throughout this section, for convenience, we use the following notation: for $\zeta \in \widetilde{{\mathbb {C}}}_+$ and $n =1,2,\ldots$, let

$$\begin{aligned} \begin{aligned}&G_n(\zeta ,x)\\&=\exp \bigg \{ \bigg [\frac{1-\zeta }{2}\bigg ]\sum _{j=1}^{n}[(e_j,x)^{\sim }]^2 + (\zeta ^{1/2 }-1) \sum _{j=1}^{n}(e_j,a)_{C_{a,b}'}(e_j,x)^{\sim } \bigg \}, \end{aligned} \end{aligned}$$

(5.1)

where $\{e_n\}_{n=1}^{\infty }$ is a complete orthonormal set of functions in $C_{a,b}'[0,T]$.

Lemma 5.1

Let k be a function in $\mathrm {Supp}_{C_{a,b}^*}[0,T]$, let $\{e_{1},\ldots ,e_{n} \}$ be an orthonormal set of functions in $C_{a,b}'[0,T]$, and let w be a function in $C_{a,b}'[0,T]$. Then for each $\zeta \in {\mathbb {C}}_+$, and $n\in {\mathbb {N}}$, the functional $\exp \big \{(w\odot k,x)^{\sim }\big \}G_n(\zeta ,x)$ is $\mu$-integrable, where $G_n$ is given by (5.1). Also, it follows that

$$\begin{aligned}&{\mathcal {W}}_n(w;k;\zeta ) \equiv \int _{C_{a,b}[0,T]}\exp \big \{ (w\odot k,x)^{\sim }\big \}G_n(\zeta ,x) d\mu (x)\nonumber \\&\quad =\zeta ^{-n/2} \exp \bigg \{ \frac{1}{2\zeta } \sum _{j=1}^{n} (e_j,w\odot k)_{C_{a,b}'}^{2} +\frac{1}{2}\bigg [\Vert w\odot k\Vert _{C_{a,b}'}^{2}-\sum _{j=1}^{n}(e_j,w \odot k)_{C_{a,b}'}^{2}\bigg ] \nonumber \\&\qquad + \zeta ^{-1/2}\sum _{j=1}^{n}(e_j,a)_{C_{a,b}'}(e_j,w\odot k)_{C_{a,b}'} \nonumber \\&\qquad + (e_{n+1}^{w\odot k},a)_{C_{a,b}'}\bigg [ \Vert w\odot k\Vert _{C_{a,b}'}^{2} -\sum _{j=1}^{n}(e_j,w\odot k)_{C_{a,b}'}^{2} \bigg ]^{1/2}\bigg \}, \end{aligned}$$

(5.2)

where

$$\begin{aligned} \begin{aligned} e_{n+1}^{w\odot k}&=\bigg [ \Vert w\odot k\Vert _{C_{a,b}'}^{2} -\sum _{j=1}^{n}(e_j,w\odot k)_{C_{a,b}'}^{2} \bigg ]^{-1/2}\\&\quad \times \bigg \{w\odot k -\sum _{j=1}^n(e_j,w\odot k)_{C_{a,b}'}e_j\bigg \}. \end{aligned} \end{aligned}$$

Proof

We note that given two functions $k\in \mathrm {Supp}_{C_{a,b}^*}[0,T]$ and $w\in C_{a,b}'[0,T]$, $w\odot k$ is an element of $C_{a,b}'[0,T]$. Using the Gram–Schmidt process, we obtain $e_{n+1}^{w\odot k}\in C_{a,b}'[0,T]$ such that $\{e_1,\ldots ,e_n, e_{n+1}^{w\odot k} \}$ forms an orthonormal set in $C_{a,b}'[0,T]$ and

$$\begin{aligned} w\odot k=\sum _{j=1}^{n}c_j e_j +c_{n+1}e_{n+1}^{w\odot k} \end{aligned}$$

where

$$\begin{aligned} c_j= {\left\{ \begin{array}{ll} (e_j,w\odot k)_{C_{a,b}'} &{},\quad j=1,\ldots ,n \\ \big [ \Vert w\odot k\Vert _{C_{a,b}'}^{2} - \sum _{j=1}^{n}(e_j,w\odot k)_{C_{a,b}'}^{2} \big ]^{1/2} &{}, \quad j=n+1 \end{array}\right. }. \end{aligned}$$

Next, for $\zeta \in {\mathbb {C}}_+$, using (5.1), (2.3), the Fubini theorem, and (2.4), it follows that

$$\begin{aligned} \begin{aligned}&\int _{C_{a,b}[0,T]}\exp \{(w\odot k,x)^{\sim }\}G_n(\zeta ,x)d\mu (x) \\&=( 2\pi )^{-(n+1)/2} \int _{{\mathbb {R}}^{n+1}} \exp \bigg \{ \bigg [ \frac{1-\zeta }{2}\bigg ] \sum \limits _{j=1}^{n}u_j^2 +(\zeta ^{1/2}-1) \sum \limits _{j=1}^{n}(e_j,a)_{C_{a,b}'}u_j \\&\qquad \qquad +\sum \limits _{j=1}^{n+1}c_j u_j - \frac{1}{2} \sum \limits _{j=1}^{n}[u_j -(e_j,a)_{C_{a,b}'} ]^2 -\frac{1}{2}[u_{n+1} -(e_{n+1}^{w\odot k},a)_{C_{a,b}'} ]^2 \bigg \}\\&\qquad \qquad \times du_1 \cdot \cdot \cdot du_n du_{n+1} \\&=\Bigg ( \prod \limits _{j=1}^{n} (2\pi )^{-1/2}\int _{{\mathbb {R}}} \exp \bigg \{ -\frac{\zeta }{2}u_j^2 +[\zeta ^{1/2}(e_j,a)_{C_{a,b}'} + c_j ]u_j \bigg \} du_j \Bigg ) \\&\quad \times \Bigg ((2\pi )^{-1/2} \int _{{\mathbb {R}}} \exp \bigg \{-\frac{1}{2}u_{n+1}^{2} + [(e_{n+1}^{w\odot k},a)_{C_{a,b}'} + c_{n+1}] u_{n+1} \bigg \} du_{n+1}\Bigg ) \\&\quad \times \exp \bigg \{ -\frac{1}{2} \sum _{j=1}^{n}(e_j,a)_{C_{a,b}'}^{2} -\frac{1}{2} (e_{n+1}^{w\odot k},a)_{C_{a,b}'}^{2} \bigg \}\\&=\zeta ^{-n/2} \exp \bigg \{\zeta ^{-1/2}\sum _{j=1}^{n}(e_j,a)_{C_{a,b}'}c_j +\frac{1}{2\zeta }\sum _{j=1}^{n}c_j^2\\&\quad + (e_{n+1}^{w\odot k},a)_{C_{a,b}'}c_{n+1} + \frac{1}{2}c_{n+1}^{2} \bigg \} \\&= \zeta ^{-n/2} \exp \bigg \{ \zeta ^{-1/2}\sum _{j=1}^{n}(e_j,a)_{C_{a,b}'}(e_j,w\odot k)_{C_{a,b}'} +\frac{1}{2\zeta } \sum _{j=1}^{n} (e_j,w\odot k)_{C_{a,b}'}^{2} \\&\quad + (e_{n+1}^{w\odot k},a)_{C_{a,b}'} \bigg [ \Vert w\odot k\Vert _{C_{a,b}'}^{2} - \sum _{j=1}^{n} (e_j,w\odot k)_{C_{a,b}'}^{2} \bigg ]^{1/ 2}\\&\quad +\frac{1}{2} \bigg [\Vert w\odot k\Vert _{C_{a,b}'}^{2} -\sum _{j=1}^{n}(e_j,w\odot k)_{C_{a,b}'}^{2}\bigg ] \bigg \} \end{aligned} \end{aligned}$$

as desired. $\square$

In our next theorem we express the analytic ${\mathcal {Z}}_k$-GFFT of functionals in ${\mathcal {E}}(C_{a,b}[0,T])$ as the limit of a sequence of ${\mathcal {Z}}_k$-function space integrals.

Theorem 5.2

Let $F\in {\mathcal {E}} (C_{a,b}[0,T])$ be given by Eq. (4.9). Given a non-zero real number q, let $\{\zeta _n \}$ be a sequence in ${\mathbb {C}}_+$ such that $\zeta _n\rightarrow -iq$. Then, for all $p\in [1,2]$, and each function k in $\mathrm {Supp}_{C_{a,b}^*}[0,T]$, it follows that

$$\begin{aligned} T_{q,k}^{(p)}(F)(y) =\lim _{n \rightarrow \infty } \zeta _{n}^{n/2} \int _{C_{a,b}[0,T]}F\big (y+{\mathcal {Z}}_{k}(x,\cdot )\big ) G_n(\zeta _{n},x) d\mu (x), \end{aligned}$$

(5.3)

for s-a.e. $y\in C_{a,b}[0,T]$, where $G_n$ is given by Eq. (5.1).

Proof

In view of Theorems 4.5 and 4.6 , it will suffice to show that Eq. (5.3) with p and F replaced with 1 and $\varPsi _w$ holds true.

From Theorem 4.5, we know that the $L_1$ analytic ${\mathcal {Z}}_k$-GFFT of $\varPsi _w$ given by (4.5), $T_{q,k}^{(1)}(\varPsi _w)$, exists. Using (4.5), (3.2), the Fubini theorem, and the first expression of (5.2) with $\zeta$ replaced with $\zeta _n$, it follows that for all $n\in {\mathbb {N}}$,

$$\begin{aligned} \begin{aligned}&\zeta _{n}^{n/2} \int _{C_{a,b}[0,T]}\varPsi _w\big (y+{\mathcal {Z}}_k(x,\cdot )\big )G_n(\zeta _{n},x)d\mu (x) \\&=\zeta _{n}^{n/2} \exp \{(w,y)^{\sim }\}\bigg [\int _{C_{a,b}[0,T]} \exp \{(w\odot k,x)^{\sim }\}G_n(\zeta _n,x)d\mu (x)\bigg ] \\&= \exp \{(w,y)^{\sim }\} \zeta _{n}^{n/2}{\mathcal {W}}_n(w;k;\zeta _n) . \end{aligned} \end{aligned}$$

(5.4)

Next, using (5.4), (5.2), Parseval’s relation, (4.7), (4.5), (4.6) with $\tau$ replaced with w, and (4.8) with $p=1$, it follows that

$$\begin{aligned}&\lim _{n \rightarrow \infty } \zeta _{n}^{n/2} \int _{C_{a,b}[0,T]}\varPsi _w\big (y+{\mathcal {Z}}_k(x,\cdot )\big )G_n(\zeta _{n},x)d\mu (x) \\&\quad = \exp \{ (w,y)^{\sim }\}\lim _{n \rightarrow \infty }\zeta _{n}^{n/2}{\mathcal {W}}_n(w;k;\zeta _n) \\&\quad = \exp \bigg \{(w,y)^{\sim } + \lim _{n \rightarrow \infty } \frac{1}{2\zeta _n} \sum _{j=1}^{n}(e_j,w\odot k)_{C_{a,b}'}^{2}\\&\qquad +\frac{1}{2} \lim _{n \rightarrow \infty }\bigg [ \Vert w\odot k\Vert _{C_{a,b}'}^{2} -\sum _{j=1}^{n}(e_j,w\odot k)_{C_{a,b}'}^{2} \bigg ] \\&\quad \quad + \lim _{n \rightarrow \infty }\zeta _n^{-1/2}\sum _{j=1}^{n}(e_j,a)_{C_{a,b}'}(e_j,w\odot k)_{C_{a,b}'} \\&\quad \quad + \lim _{n \rightarrow \infty } (e_{n+1},a)_{C_{a,b}'}\bigg [ \Vert w\odot k\Vert _{C_{a,b}'}^{2} -\sum _{j=1}^{n}(e_j,w\odot k)_{C_{a,b}'}^{2} \bigg ]^{1/2}\bigg \} \\&\quad = \exp \{ (w,y)^{\sim }\}K^a_{q,w, k}\\&\quad = \varPsi _w(y)K^a_{q,w, k}\\&\quad =\varPsi _{w}^{q,\tau ,k} (y)\\&\quad =T_{q,k}^{(1)}(\varPsi _w)(y) \end{aligned}$$

for s-a.e. $y\in C_{a,b}[0,T]$, as desired. $\square$

The following corollary follows immediately from (4.4) and (5.3).

Corollary 5.3

Let F, q and $\{\zeta _n \}$ be as in Theorem 5.2. Then, for each function k in $\mathrm {Supp}_{C_{a,b}^*}[0,T]$, it follows that

$$\begin{aligned} I_k^{\mathrm{anf}_{q}}[F] =\lim _{n \rightarrow \infty } \zeta _{n}^{n/2} \int _{C_{a,b}[0,T]} F\big ({\mathcal {Z}}_k (x,\cdot )\big )G_n(\zeta _{n},x)d\mu (x), \end{aligned}$$

where $G_n$ is given by Eq. (5.1).

We establish our next corollary after a careful examination of the proof of Theorem 5.2, and by using Eq. (4.1) instead of (4.4).

Corollary 5.4

Let $F\in {\mathcal {E}} (C_{a,b}[0,T])$ be given by Eq. (4.9). Let $\lambda \in {\mathbb {C}}_+$, and let $\{\zeta _n \}$ be a sequence in ${\mathbb {C}}_+$ such that $\zeta _n\rightarrow \lambda$. Then, for each function k in $\mathrm {Supp}_{C_{a,b}^*}[0,T]$, it follows that

$$\begin{aligned} I_k^{\mathrm{an}_{\lambda }}[F ] =\lim _{n \rightarrow \infty } \zeta _{n}^{n/2} \int _{C_{a,b}[0,T]} F\big ({\mathcal {Z}}_k (x,\cdot )\big )G_n(\zeta _{n},x)d\mu (x), \end{aligned}$$

(5.5)

where $G_n$ is given by Eq. (5.1).

Our next result, namely a change of scale formula for function space integrals, now follows easily by Corollary 5.4 above.

Corollary 5.5

Let $F\in {\mathcal {E}} (C_{a,b}[0,T])$ be given by Eq. (4.9). Then for any $\rho > 0$, and each function k in $\mathrm {Supp}_{C_{a,b}^*}[0,T]$, it follows that

$$\begin{aligned} \int _{C_{a,b}[0,T]} F(\rho {\mathcal {Z}}_k (x,\cdot ))d\mu (x) = \lim _{n \rightarrow \infty } \rho ^{-n} \int _{C_{a,b}[0,T]} F({\mathcal {Z}}_k (x,\cdot ))G_n(\rho ^{-2},x)d\mu (x), \end{aligned}$$

(5.6)

where $G_n$ is given by Eq. (5.1).

Proof

Note that for every $F\in {\mathcal {E}}(C_{a,b}[0,T])$ and all $\rho >0$, the function space integral in the left-hand side of (5.6) exists. To ensure the equality in (5.6), simply choose $\lambda = \rho ^{-2}$ and $\zeta _n =\rho ^{-2}$ for every $n\in {\mathbb {N}}$ in (5.5). $\square$

6 Concluding Remark

It is known that the class ${\mathcal {E}}(C_{a,b}[0,T])$ is a dense subspace of the space $L_2(C_{a,b}[0,T])$. For a related work, see [9, 11]. Thus, using the $L_2$-approximation [11, Remark 4], one can develop the sequential approximation such as Eq. (5.3) for functionals F in $L_2(C_{a,b}[0,T])$ whose $L_p$ analytic ${\mathcal {Z}}_k$-GFFT $T_{q,k}^{(p)}(F)$ exists. But, there exists a (bounded) functional F in $L_2(C_{a,b}[0,T])$ whose $L_p$ ${\mathcal {Z}}_k$-GFFT $T_{q,k}^{(p)}(F)$ does not exist, see [5].

Indeed, the class ${\mathcal {E}}(C_{a,b}[0,T])$ is a very rich class of functionals on $C_{a,b}[0,T]$. It contains many meaningful functionals which discussed in quantum mechanics. We finish this paper with a very simple example for such functionals, which arises in quantum mechanics.

Example 6.1

Consider the functional $F_S$ given by

$$\begin{aligned} F_S(x)=\exp \bigg \{\int _0^T x(t)d b(t)\bigg \} \end{aligned}$$

for s-a.e. $x\in C_{a,b}[0,T]$. Then $F_S$ is an element of ${\mathcal {E}}(C_{a,b}[0,T])$ because

$$\begin{aligned} F_S(x)=\exp \big \{(Sb,x)^{\sim })\big \} \end{aligned}$$

where $S: C_{a,b}'[0,T]\rightarrow C_{a,b}'[0,T]$ is an operator defined by

$$\begin{aligned} Sw(t)=\int _0^t[w(T)-w(s)]db(s). \end{aligned}$$

One can see that the adjoint operator $S^*$ of S is given by

$$\begin{aligned} S^*w(t)=\int _0^t w(s)db(s). \end{aligned}$$