Partial Derivative Approach to the Integral Transform for the Function Space in the Banach Algebra

Abstract

We investigate some relationships among the integral transform, the function space integral and the first variation of the partial derivative approach in the Banach algebra defined on the function space. We prove that the function space integral and the integral transform of the partial derivative in some Banach algebra can be expanded as the limit of a sequence of function space integrals.

1. Introduction

The first variation defined by the partial derivative approach was defined in [1]. Relationships among the Function space integral and transformations and translations were developed in [2,3,4]. Integral transforms for the function space were expanded upon in [5,6,7,8,9].

A change of scale formula and a scale factor for the Wiener integral were expanded in [10,11,12] and in [13] and in [14].

Relationships among the function space integral and the integral transform and the first variation were expanded in [13,15,16] and in [17,18]

In this paper, we expand those relationships among the function space integral, the integral transform and the first variation into the Banach algebra [19].

2. Preliminaries

Let $C_0[0,T]$ be the class of real-valued continuous functions x on $[0,T]$ with $x\left(0\right)=0$, which is a function space. Let M denote the class of all Wiener measurable subsets of $C_0[0,T]$ and let m denote the Wiener measure. Then $(C_0[0,T],M,m)$ is a complete measure space and

$$E_x\left[F\left(x\right)\right]={\int }_{C_0[0,T]}F\left(x\right)dm\left(x\right)$$

is called the Wiener integral of a function F defined on the function space $C_0[0,T]$.

A subset E of $C_0[0,T]$ is said to be scale-invariant measurable provided $\rho E\in M$ for all $\rho >0$ and a scale invariant measurable set N is said to be scale-invariant null provided $m\left(\rho N\right)=0$ for each $\rho >0$. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s-a.e.). If two functions F and G are equal s-a.e., we write $F\approx G$.

Definition 1.

For the definition of the analytic Wiener integral and the analytic Feynman integral, see Definition 1 in [18]: ${\mathit{C}}_{+}=\left\{\lambda \right|Re\left(\lambda \right)>0\}$ and ${\mathit{C}}_{+}^{\sim }=\left\{\lambda \right|Re\left(\lambda \right)\ge 0\}$. For real $\lambda >0$,

$$J_F\left(\lambda \right)=E_x\left(F\left({\lambda }^{-\frac{1}{2}}x\right)\right).$$

(1)

For each $z\in {\mathit{C}}_{+}$, the analytic Wiener integral is defined by

$$E_x^{an{w}_z}\left[F\left(x\right)\right]=E_x\left(F\left(z^{-\frac{1}{2}}x\right)\right)=J_F^{*}\left(z\right).$$

(2)

Whenever $z\to -iq$ through ${\mathit{C}}_{+}$, the analytic Feynman integral is defined by

$$E_x^{an{f}_q}\left[F\left(x\right)\right]=\underset{z\to -iq}{lim}{E}_x^{an{w}_z}\left[F\left(x\right)\right],$$

(3)

where $i^2=-1$.

Notation 1.

For $\lambda \in {\mathit{C}}_{+}$ and for $s-a.e.y\in C_0[0,T]$, let

$$\left(T_{\lambda }\left(F\right)\right)\left(y\right)=E_x^{an{w}_{\lambda }}\left[F(x+y)\right].$$

(4)

Definition 2.

For the $L_1$-analytic Fourier–Feynmann transform, see Definition 2 in [5]:

$$\left(T_q^{\left(1\right)}\left(F\right)\right)\left(y\right)=\underset{\lambda \to -iq}{lim}{E}_x^{an{w}_{\lambda }}\left[F(x+y)\right]=E_x^{an{f}_q}\left[F(x+y)\right],$$

(5)

whenever $\lambda \to -iq$ through ${\mathit{C}}_{+}$ (if it exists). See [5,9].

Definition 3

(Ref. [1]). The first variation of a Wiener measurable functional F in the direction $w\in C_0[0,T]$ which is defined by the partial derivative as

$$\delta F\left(x\right|w)=\frac{\partial }{\partial h}{F(x+hw)|}_{h=0}.$$

(6)

We will denote it by $[D,F,x,w]$.

Remark 1.

For $a\in {\mathit{C}}_{+}$ and $b\in R$,

$$\begin{array}{ccc}& & {\int }_{R}exp\{-a{u}^2+ibu\}du=\sqrt{\frac{\pi }{a}}exp\{-\frac{b^2}{4a}\}.\hfill \end{array}$$

(7)

3. Results (1). On ${\mathit{C}}_{\mathbf{0}}[\mathbf{0},\mathit{T}]$

Let

$$\begin{array}{ccc}& & F\left(x\right)={\int }_{L_2[0,T]}exp\left\{i[I,v\left(t\right),x\left(t\right)]\right\}df\left(v\right)\hfill \end{array}$$

(8)

in some Banach algebra $\mathbf{S}$ defined on $C_0[0,T]$ in [19], where $[I,v\left(t\right),x\left(t\right)]={\int }_0^{T}v\left(t\right)dx\left(t\right)$ and assume that ${\int }_{L_2[0,T]}{\left|\right|v\left|\right|}_{2}d\left|f\right|\left(v\right)<\infty$.

Suppose that formulas in this section hold for $s-a.e.w\in C_0[0,T]$ and for $s-a.e.y\in C_0[0,T]$.

Lemma 1.

$$\begin{array}{ccc}& & [D,F,x,w]={\int }_{L_2[0,T]}\left(i[I,v\left(t\right),w\left(t\right)]\right)exp\left\{i[I,v\left(t\right),x\left(t\right)]\right\}df\left(v\right),\hfill \end{array}$$

(9)

where $[I,v\left(t\right),w\left(t\right)]={\int }_0^{T}v\left(t\right)dw\left(t\right).$

Proof .

By Equation (6).

$$\begin{array}{ccc}& & [D,F,x,w]\hfill \\ & =& \frac{\partial }{\partial h}{F(x+hw)|}_{h=0}\hfill \\ & =& \frac{\partial }{\partial h}{\int }_{L_2[0,T]}{exp\{ih[I,v\left(t\right),w\left(t\right)]+i[I,v\left(t\right),x\left(t\right)]\}df\left(v\right)|}_{h=0}\hfill \\ & =& {\int }_{L_2[0,T]}\left(i[I,v\left(t\right),w\left(t\right)]\right)exp\left\{i[I,v\left(t\right),x\left(t\right)]\right\}df\left(v\right).\hfill \end{array}$$

(10)

and

$$\begin{array}{ccc}& & \left|E_w\right({\int }_{L_2[0,T]}[I,v\left(t\right),w\left(t\right)]·exp\left\{i[I,v\left(t\right),x\left(t\right)]\right\}df\left(v\right)\left)\right|\hfill \\ & \le & E_w\left({\int }_{L_2[0,T]}\right|[I,v\left(t\right),w\left(t\right)]\left|d\left|f\right|\left(v\right)\right).\hfill \end{array}$$

(11)

Then

$$\begin{array}{ccc}& & E_w\left({\int }_{L_2[0,T]}\right|[I,v\left(t\right),w\left(t\right)]\left|d\left|f\right|\left(v\right)\right)\hfill \\ & =& {\int }_{L_2[0,T]}{E}_w\left(\right|[I,v\left(t\right),w\left(t\right)]\left|\right)d\left|f\right|\left(v\right)\hfill \\ & =& {\int }_{L_2[0,T]}[\frac{1}{\sqrt{{2\pi \left|v\right||}_2^2}}{\int }_{-\infty }^{+\infty }\left|u\right|exp\{-\frac{u^2}{{2\left|\right|v\left|\right|}_2^2}\}du]d\left|f\right|\left(v\right)\hfill \\ & =& \sqrt{\frac{2}{\pi }}{\int }_{L_2[0,T]}{\left|\right|v\left|\right|}_{2}d\left|f\right|\left(v\right)<\infty \hfill \end{array}$$

(12)

where $E_w\left(F\left(w\right)\right)={\int }_{C_0[0,T]}F\left(w\right)dm\left(w\right).$ So,

$${\int }_{L_2[0,T]}\left|[I,v\left(t\right),w\left(t\right)]\right|d\left|f\right|\left(v\right)<\infty .$$

Therefore, $[D,F,x,w]$ exists.  ☐

Theorem 1.

$$\begin{array}{ccc}& \left[1\right].& E_x^{an{w}_z}\left([D,F,x+y,w]\right)\hfill \\ & & ={\int }_{L_2[0,T]}\left(i[I,v\left(t\right),w\left(t\right)]\right)exp\{-\frac{1}{2z}{\int }_0^T{v}^2\left(s\right)ds+i[I,v\left(t\right),y\left(t\right)]\}df\left(v\right)\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& \left[2\right].& E_x^{an{f}_q}\left([D,F,x+y,w]\right)\hfill \\ & & ={\int }_{L_2[0,T]}\left(i[I,v\left(t\right),w\left(t\right)]\right)exp\{-\frac{i}{2q}{\int }_0^T{v}^2\left(s\right)ds+i[I,v\left(t\right),y\left(t\right)]\}df\left(v\right),\hfill \end{array}$$

(14)

where $[D,F,x+y,w]=\delta F(x+y|w)$.

Proof. 

[1]. For $z\in {\mathbf{C}}_{+}$,

$$\begin{array}{ccc}& & E_x^{an{w}_z}\left([D,F,x+y,w]\right)\hfill \\ & =& E_x\left([D,F,z^{-\frac{1}{2}}x,y,w]\right)\hfill \\ & =& E_x\left({\int }_{L_2[0,T]}\right(i[I,v\left(t\right),w\left(t\right)])\hfill \\ & & ·exp\{i{z}^{-\frac{1}{2}}[I,v\left(t\right),x\left(t\right)]+i[I,v\left(t\right),y\left(t\right)]\}df\left(v\right))\hfill \\ & =& {\int }_{L_2[0,T]}\left(i[I,v\left(t\right),w\left(t\right)]\right)\hfill \\ & & ·E_x(exp\{i{z}^{-\frac{1}{2}}[I,v\left(t\right),x\left(t\right)]\left\}\right)·exp\left\{i[I,v\left(t\right),y\left(t\right)]\right\}df\left(v\right)\hfill \\ & =& {\int }_{L_2[0,T]}\left(i[I,v\left(t\right),w\left(t\right)]\right)·exp\{-\frac{1}{2z}{\left|\right|v\left|\right|}_2^2+i[I,v\left(t\right),y\left(t\right)]\}df\left(v\right).\hfill \end{array}$$

(15)

[2].

$$\begin{array}{ccc}& & E_x^{an{f}_q}\left([D,F,x+y,w]\right)\hfill \\ & =& \underset{z\to -iq}{lim}{E}_x^{an{w}_z}\left([D,F,x+y,w]\right)\hfill \\ & =& \underset{z\to -iq}{lim}{\int }_{L_2[0,T]}\left(i[I,v\left(t\right),w\left(t\right)]\right)·exp\left\{\right(-\frac{1}{2z}{\left|\right|v\left|\right|}_2^2+i[I,v\left(t\right),y\left(t\right)]\}df\left(v\right)\hfill \\ & =& {\int }_{L_2[0,T]}\left(i[I,v\left(t\right),w\left(t\right)]\right)·exp\{-\frac{i}{2q}\left[{\int }_0^T{v}^2\left(s\right)ds\right]+i[I,v\left(t\right),y\left(t\right)]\}df\left(v\right).\hfill \end{array}$$

(16)

 ☐

Lemma 2.

For $\lambda \in {\mathit{C}}_{+}$,

$$\begin{array}{ccc}& & exp\left\{\frac{1-\lambda }{2}\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\right\}·[D,F,x+y,w]\hfill \end{array}$$

(17)

is a Wiener integrable function of $x\in C_0[0,T]$.

Proof. 

$$\begin{array}{ccc}& & E_x(exp\{\frac{1-\lambda }{2}\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\left\}[D,F,x+y,w]\right)\hfill \\ & =& E_x(exp\{\frac{1-\lambda }{2}\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\}\hfill \\ & & ·\left[{\int }_{L_2[0,T]}\right(i[I,v\left(t\right),w\left(t\right)])·exp\{i[I,v\left(t\right),x\left(t\right)]+i[I,v\left(t\right),y\left(t\right)]\left\}df\left(v\right)\right])\hfill \\ & =& {\int }_{L_2[0,T]}\left(i[I,v\left(t\right),w\left(t\right)]\right)·E_x(exp\{\frac{1-\lambda }{2}\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\}\hfill \\ & & ·exp\left\{i\right[I,v\left(t\right),x\left(t\right)]+i[I,v\left(t\right),y\left(t\right)\left]\right\}\left)df\right(v).\hfill \end{array}$$

(18)

and

$$\begin{array}{ccc}& & E_x(exp\{\frac{1-\lambda }{2}\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\}·exp\{i[I,v\left(t\right),x\left(t\right)]+i[I,v\left(t\right),y\left(t\right)]\left\}\right)\hfill \\ & \le & E_x(exp\{\frac{1-\lambda }{2}\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\left\}\right)\hfill \\ & =& {\left(2\pi \right)}^{-\frac{n}{2}}{\int }_{R^n}exp\{-\frac{\lambda }{2}\sum _{k=1}^n{u}_k^2\}d\overrightarrow{u}\hfill \\ & =& {\left(2\pi \right)}^{-\frac{n}{2}}·{\left(\frac{2\pi }{\lambda }\right)}^{\frac{n}{2}}\hfill \\ & =& {\lambda }^{-\frac{n}{2}}.\hfill \end{array}$$

(19)

Therefore,

$$\begin{array}{ccc}& & \left|{\int }_{L_2[0,T]}\right(i[I,v\left(t\right),w\left(t\right)])·E_x(exp\left\{\frac{1-\lambda }{2}\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\right\}\hfill \\ & & ·exp\left\{i\right[I,v\left(t\right),x\left(t\right)]+i[I,v\left(t\right),y\left(t\right)\left]\right\}\left)df\right(v\left)\right|\hfill \\ & \le & {\lambda }^{-\frac{n}{2}}·{\int }_{L_2[0,T]}\left|[I,v\left(t\right),w\left(t\right)]\right|d\left|f\right|\left(v\right).\hfill \end{array}$$

(20)

By the Wiener integration theorem,

$$\begin{array}{ccc}& & E_w\left({\int }_{L_2[0,T]}\right|[I,v\left(t\right),w\left(t\right)]\left|d\left|f\right|\left(v\right)\right)\hfill \\ & =& {\int }_{L_2[0,T]}[\frac{1}{\sqrt{{2\pi \left|\right|v\left|\right|}_2^2}}·{\int }_{-\infty }^{+\infty }\left|u\right|·exp\{-\frac{u^2}{{2\left|\right|v\left|\right|}_2^2}\left\}du\right]d\left|f\right|\left(v\right)\hfill \\ & =& {\left(\frac{2}{\pi }\right)}^{\frac{1}{2}}{\int }_{L_2[0,T]}{\left|\right|v\left|\right|}_{2}d\left|f\right|\left(v\right)<\infty .\hfill \end{array}$$

(21)

 ☐

Lemma 3

(Ref. [12]). Let ${\left\{{\varphi }_j\right\}}_{j=1}^m$ be an orthonormal set in $L_2[0,T]$. Then for $v\in L_2[0,T]$ and for $\lambda \in {\mathit{C}}_{+}$,

$$\begin{array}{ccc}& & E_x(exp\{\frac{1-\lambda }{2}\sum _{k=1}^m{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2+i[I,v\left(t\right),x\left(t\right)]\left\}\right)\hfill \\ & =& {\lambda }^{-\frac{m}{2}}·exp\{-\frac{1-\lambda }{2\lambda }\sum _{k=1}^m{\left({\int }_0^T{\varphi }_k\left(s\right)v\left(s\right)ds\right)}^2-\frac{1}{2}{\int }_0^T{v}^2\left(s\right)ds\left]\right\}.\hfill \end{array}$$

(22)

Theorem 2.

For $z\in {\mathit{C}}_{+}$,

$$\begin{array}{ccc}& & E_x^{an{w}_z}\left([D,F,x+y,w]\right)\hfill \\ & =& \underset{n\to \infty }{lim}{z}^{\frac{n}{2}}·E_x(exp\{\frac{1-z}{2}\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\left\}[D,F,x+y,w]\right)\hfill \end{array}$$

(23)

Proof by Lemma 1.

$$\begin{array}{ccc}& & [D,F,z^{-\frac{1}{2}}x+y,w]\hfill \\ & =& {\int }_{L_2[0,T]}\left(i[I,v\left(t\right),w\left(t\right)]\right)exp\{i{z}^{-\frac{1}{2}}[I,v\left(t\right),x\left(t\right)]+i[I,v\left(t\right),y\left(t\right)]\}df\left(v\right).\hfill \end{array}$$

(24)

By Lemma 3,

$$\begin{array}{ccc}& & \underset{n\to \infty }{lim}{z}^{\frac{n}{2}}·E_x(exp\{\frac{1-z}{2}\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\left\}[D,F,x+y,w]\right)\hfill \\ & =& \underset{n\to \infty }{lim}{z}^{\frac{n}{2}}{\int }_{L_2[0,T]}{E}_x(exp\{\frac{1-z}{2}\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2+i[I,v\left(t\right),x\left(t\right)]\left\}\right)\hfill \\ & & ·\left(i\right[I,v\left(t\right),w\left(t\right)\left]\right)·exp\left\{i\right[I,v\left(t\right),y\left(t\right)\left]\right\}df\left(v\right)\hfill \\ & =& \underset{n\to \infty }{lim}{z}^{\frac{n}{2}}{\int }_{L_2[0,T]}{z}^{-\frac{n}{2}}exp\{\frac{z-1}{2z}\sum _{k=1}^m{\int }_0^T{\varphi }_k\left(s\right)v\left(s\right),ds{]}^2-\frac{1}{2}{\int }_0^T{v}^2\left(t\right)dt]\}\hfill \\ & & ·\left(i\right[I,v\left(t\right),w\left(t\right)\left]\right)·exp\left\{i\right[I,v\left(t\right),y\left(t\right)\left]\right\}df\left(v\right)\hfill \\ & =& \underset{n\to \infty }{lim}{\int }_{L_2[0,T]}\left(i[I,v\left(t\right),w\left(t\right)]\right)exp\{-\frac{1}{2z}\sum _{k=1}^n{\left[{\int }_0^T{\varphi }_k\left(s\right)v\left(s\right)ds\right]}^2\}\hfill \\ & & ·exp\{\frac{1}{2}\sum _{k=1}^n{\left[{\int }_0^T{\varphi }_k\left(s\right)v\left(s\right)ds\right]}^2-\frac{1}{2}{\int }_0^T{v}^2\left(t\right)dt\}\hfill \\ & & ·exp\left\{i\right[I,v\left(t\right),y\left(t\right)\left]\right\}df\left(v\right)\hfill \\ & =& {\int }_{L_2[0,T]}\left(i[I,v\left(t\right),w\left(t\right)]\right)exp\{-\frac{1}{2z}{\left|\right|v\left|\right|}_2^2+[I,v\left(t\right),y\left(t\right)]\}df\left(v\right)\hfill \\ & =& {\int }_{L_2[0,T]}\left(i[I,v\left(t\right),w\left(t\right)]\right)·E_x(exp\{i{z}^{-\frac{1}{2}}[I,v\left(t\right),x\left(t\right)]\left\}\right)\hfill \\ & & ·exp\left\{i\right[I,v\left(t\right),y\left(t\right)\left]\right\}df\left(v\right)\hfill \\ & =& E_x\left({\int }_{L_2[0,T]}\right(i[I,v\left(t\right),w\left(t\right)])\hfill \\ & & ·exp\{i{z}^{-\frac{1}{2}}[I,v\left(t\right),x\left(t\right)]+i[I,v\left(t\right),y\left(t\right)]\}df\left(v\right))\hfill \\ & =& E_x\left([D,F,z^{-\frac{1}{2}}x+y,w]\right)\hfill \\ & =& E_x^{an{w}_z}\left([D,F,x+y,w]\right).\hfill \end{array}$$

(25)

 ☐

Theorem 3.

For real $\rho >0$,

$$\begin{array}{ccc}& & E_x\left([D,F,\rho x+y,w\right])\hfill \\ & =& \underset{n\to \infty }{lim}{\rho }^{-n}·E_x(exp\{\frac{{\rho }^2-1}{2{\rho }^2}\sum _{k=1}^m{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\left\}[D,F,x+y,w]\right).\hfill \end{array}$$

(26)

Proof. 

For real $\lambda >0$,

$$\begin{array}{ccc}& & E_x^{an{w}_{\lambda }}\left([D,F,x+y,w]\right)\hfill \\ & \equiv & E_x\left([D,F,{\lambda }^{-\frac{1}{2}}x+y,w]\right)\hfill \\ & =& \underset{n\to \infty }{lim}{\lambda }^{\frac{n}{2}}·E_x(exp\{\frac{1-\lambda }{2}\sum _{k=1}^m{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\left\}[D,F,x+y,w]\right).\hfill \end{array}$$

(27)

Taking $\lambda ={\rho }^{-2}$, we have the result.  ☐

Theorem 4.

$$\begin{array}{ccc}& & E_x^{an{f}_q}\left([D,F,x+y,w]\right)\hfill \\ & =& \underset{n\to \infty }{lim}{{\lambda }_n}^{\frac{n}{2}}·E_x(exp\{\frac{1-{\lambda }_n}{2}\sum _{k=1}^m{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\left\}[D,F,x+y,w]\right),\hfill \end{array}$$

(28)

whenever $\left\{{\lambda }_n\right\}\to -iq$ through $C_{+}$.

Proof .

by Theorem 2,

$$\begin{array}{ccc}& & E_x^{an{f}_q}\left([D,F,x+y,w]\right)\hfill \\ & =& \underset{n\to \infty }{lim}{E}_x^{an{w}_{{\lambda }_n}}\left([D,F,x+y,w]\right)\hfill \\ & =& \underset{n\to \infty }{lim}{{\lambda }_n}^{\frac{n}{2}}·E_x(exp\{\frac{1-{\lambda }_n}{2}\sum _{k=1}^m{[I,{\varphi }_k\left(t\right),x\left(t\right)]}^2\left\}[D,F,x+y,w]\right).\hfill \end{array}$$

(29)

 ☐

4. Results (2). on ${\mathit{C}}_{\mathbf{0}}^{\mathbf{\nu }}[\mathbf{0},\mathit{T}]$

In this section, we expand the result about the function:

$$\begin{array}{ccc}& & F\left(\overrightarrow{x}\right)={\int }_{L_2^{\nu }[0,T]}exp\left\{i\sum _{j=1}^{\nu }[I,v_j\left(t\right),x_j\left(t\right)]\right\}df\left(\overrightarrow{v}\right)\hfill \end{array}$$

(30)

in some Banach algebra ${\mathbf{S}}^{\prime }$ defined on $C_0^{\nu }[0,T]$ in [14].

Let $\overrightarrow{w}=(w_1,\cdots ,w_{\nu })$, where $w_j\in C_0[0,T]$ is absolutely continuous on $[0,T]$ and $w_j^{{}^{\prime }}\left(t\right)\in L_2[0,T]$ for $1\le j\le \nu$. Suppose also that $M={Max}_{1\le j\le \nu }\left|\right|w_j^{{}^{\prime }}{\left|\right|}_2<\infty$ and we assume that ${\int }_{L_2^{\nu }[0,T]}{\sum }_{j=1}^{\nu }\left|\right|v_j{\left|\right|}_{2}d\left|f\right|\left(\overrightarrow{v}\right)<\infty$.

Suppose that formulas in this section hold $s-a.e.\overrightarrow{w}\in C_0^{\nu }[0,T]$ and for $s-a.\overrightarrow{y}\in C_0^{\nu }[0,T]$.

Lemma 4.

$$\begin{array}{ccc}& & [D,F,\overrightarrow{x},\overrightarrow{w}]={\int }_{L_2^{\nu }[0,T]}\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)exp\left\{i\sum _{j=1}^{\nu }[I,v_j\left(t\right),x_j\left(t\right)]\right\}df\left(\overrightarrow{v}\right).\hfill \end{array}$$

(31)

Proof .

By Equation (6).

$$\begin{array}{ccc}& & [D,F,\overrightarrow{x},\overrightarrow{w}]\hfill \\ & =& \frac{\partial }{\partial h}F(\overrightarrow{x}+h\overrightarrow{w}){|}_{h=0}\hfill \\ & =& \frac{\partial }{\partial h}{\int }_{L_2^{\nu }[0,T]}exp\{ih\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]+i\sum _{j=1}^{\nu }[I,v_j\left(t\right),x_j\left(t\right)]\}df\left(\overrightarrow{v}\right){|}_{h=0}\hfill \\ & =& {\int }_{L_2^{\nu }[0,T]}\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)exp\left\{i\sum _{j=1}^{\nu }[I,v_j\left(t\right),x_j\left(t\right)]\right\}df\left(\overrightarrow{v}\right).\hfill \end{array}$$

(32)

We know that the Paley–Wiener–Zygmund integral equals to the Riemann–Stieltzes integral

$${\int }_0^{T}f\left(t\right)dg\left(t\right)={\int }_0^{T}f\left(t\right)g^{\prime }\left(t\right)dt,$$

if g is absolutely continuous in $[0,T]$ with $g^{\prime }\left(t\right)\in L_2[0,T]$.

For $1\le j\le \nu$, ${\int }_0^T{v}_j\left(t\right)d{w}_j\left(t\right)={\int }_0^T{v}_j\left(t\right)w_j^{{}^{\prime }}\left(t\right)dt$. Therefore

$$\begin{array}{ccc}& & \left|{\int }_{L_2^{\nu }[0,T]}\right(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)])exp\{i\sum _{j=1}^{\nu }[I,v_j\left(t\right),x_j\left(t\right)]\left\}df\left(\overrightarrow{v}\right)\right|\hfill \\ & \le & {\int }_{L_2^{\nu }[0,T]}\left|\sum _{j=1}^{\nu }{\int }_0^T{v}_j\left(t\right)w_j^{{}^{\prime }}\left(t\right)dt\right|d\left|f\right|\left(\overrightarrow{v}\right)\hfill \\ & \le & {\int }_{L_2^{\nu }[0,T]}\sum _{j=1}^{\nu }\left|\right|v_j{\left|\right|}_2·\left|\right|w_j^{{}^{\prime }}{\left|\right|}_{2}d\left|f\right|\left(\overrightarrow{v}\right)\hfill \\ & \le & {\int }_{L_2^{\nu }[0,T]}\sum _{j=1}^{\nu }\left|\right|v_j{\left|\right|}_2·[{Max}_{1\le j\le \nu }\left|\right|w_j^{{}^{\prime }}{\left|\right|}_2\left]d\right|f|\left(\overrightarrow{v}\right)\hfill \\ & =& M·{\int }_{L_2^{\nu }[0,T]}\sum _{j=1}^{\nu }\left|\right|v_j{\left|\right|}_{2}d\left|f\right|\left(\overrightarrow{v}\right)\hfill \\ & <& \infty ,\hfill \end{array}$$

(33)

where $M={Max}_{1\le j\le \nu }\left|\right|w_j^{{}^{\prime }}{\left|\right|}_2<\infty$.  ☐

Theorem 5.

$$\begin{array}{cc}\hfill \left(1\right).& E_{\overrightarrow{x}}^{an{w}_z}\left([D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right)\hfill \\ \hfill =& {\int }_{L_2^{\nu }[0,T]}\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)exp\{-\frac{1}{2z}\sum _{j=1}^{\nu }{\int }_0^T{v}_j^2\left(s\right)ds+i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\}df\left(\overrightarrow{v}\right).\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill \left(2\right).& E_{\overrightarrow{x}}^{an{f}_q}([D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}])\hfill \\ \hfill =& {\int }_{L_2^{\nu }[0,T]}\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)exp\{-\frac{i}{2q}\sum _{j=1}^{\nu }{\int }_0^T{v}_j^2\left(s\right)ds+i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\}df\left(\overrightarrow{v}\right).\hfill \end{array}$$

(35)

Proof .

$\left(1\right)$. For $z\in {\mathbf{C}}_{+}$,

$$\begin{array}{ccc}& & E_{\overrightarrow{x}}^{an{w}_z}\left([D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right)\hfill \\ & =& E_{\overrightarrow{x}}\left([D,F,z^{-\frac{1}{2}}\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right)\hfill \\ & =& E_{\overrightarrow{x}}\left({\int }_{L_2^{\nu }[0,T]}\right(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)])\hfill \\ & & ·exp\{i{z}^{-\frac{1}{2}}[I,v_j\left(t\right),x_j\left(t\right)]+i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\}df\left(\overrightarrow{v}\right))\hfill \\ & =& {\int }_{L_2^{\nu }[0,T]}\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)·\left[E_{\overrightarrow{x}}(exp\{i{z}^{-\frac{1}{2}}\sum _{j=1}^{\nu }[I,v_j\left(t\right),x_j\left(t\right)]\right\})\hfill \\ & & ·exp\left\{i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\right\}df\left(\overrightarrow{v}\right)\hfill \\ & =& {\int }_{L_2^{\nu }[0,T]}\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)·exp\{-\frac{1}{2z}\sum _{j=1}^{\nu }\left|\right|v_j{\left|\right|}_2^2\}\hfill \\ & & ·exp\left\{i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\right\}df\left(\overrightarrow{v}\right)\hfill \\ & =& {\int }_{L_2^{\nu }[0,T]}\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)exp\{-\frac{1}{2z}\sum _{j=1}^{\nu }\left|\right|v_j{\left|\right|}_2^2+i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\}df\left(\overrightarrow{v}\right)\hfill \end{array}$$

(36)

$\left(2\right).$

$$\begin{array}{ccc}& & E_{\overrightarrow{x}}^{an{f}_q}([D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}])\hfill \\ & =& \underset{z\to -iq}{lim}{E}_{\overrightarrow{x}}^{an{w}_z}\left([D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right)\hfill \\ & =& \underset{z\to -iq}{lim}{\int }_{L_2^{\nu }[0,T]}\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)·exp\{-\frac{1}{2z}\sum _{j=1}^{\nu }\left|\right|v_j{\left|\right|}_2^2+i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\}df\left(\overrightarrow{v}\right)\hfill \\ & =& {\int }_{L_2^{\nu }[0,T]}\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)·exp\{-\frac{i}{2q}\sum _{j=1}^{\nu }{\int }_0^T{v}_j^2\left(s\right)ds+i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\}df\left(\overrightarrow{v}\right)\hfill \end{array}$$

(37)

 ☐

Lemma 5.

For $\lambda \in {\mathit{C}}_{+}$,

$$\begin{array}{ccc}& & exp\left\{\frac{1-\lambda }{2}\sum _{j=1}^{\nu }\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\right\}[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\hfill \end{array}$$

(38)

is a Wiener-integrable function of $\overrightarrow{x}\in C_0^{\nu }[0,T]$.

Proof. 

First we have

$$\begin{array}{ccc}& & E_{\overrightarrow{x}}(exp\{\frac{1-\lambda }{2}\sum _{j=1}^{\nu }\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\left\}[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right)\hfill \\ & =& E_{\overrightarrow{x}}(exp\{\frac{1-\lambda }{2}\sum _{k=1}^n(\sum _{j=1}^{\nu }{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\}\hfill \\ & & ·\left[{\int }_{L_2^{\nu }[0,T]}\right(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)])\hfill \\ & & ·exp\{i\sum _{j=1}^{\nu }[I,v_j\left(t\right),x\left(t\right)]+i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\}df\left(\overrightarrow{v}\right))\hfill \\ & =& {\int }_{L_2^{\nu }[0,T]}\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)·E_{\overrightarrow{x}}(exp\{\frac{1-\lambda }{2}\sum _{j=1}^{\nu }\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\}\hfill \\ & & ·exp\{i\sum _{j=1}^{\nu }[I,v_j\left(t\right),x_j\left(t\right)]+i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\})df\left(\overrightarrow{v}\right).\hfill \end{array}$$

(39)

Therefore we have

$$\begin{array}{ccc}& & \left|E_{\overrightarrow{x}}\right(exp\left\{\frac{1-\lambda }{2}\sum _{j=1}^{\nu }\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\right\}[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\left)\right|\hfill \\ & \le & {\int }_{L_2^{\nu }[0,T]}\left|\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right|·E_{\overrightarrow{x}}\left(\right|exp\left\{\frac{1-\lambda }{2}\sum _{j=1}^{\nu }\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\right\}\left|\right)d\left|f\right|\left(\overrightarrow{v}\right)\hfill \\ & \le & {\int }_{L_2^{\nu }[0,T]}\left|\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right|·|{\left(2\pi \right)}^{-\frac{n}{2}}{\int }_{R^{\mathbf{\nu }n}}exp\{-\frac{\lambda }{2}\sum _{j=1}^{\nu }\sum _{k=1}^n{u}_{j,k}^2\left\}d\overrightarrow{u}\right|d\left|f\right|\left(\overrightarrow{v}\right)\hfill \\ & =& {\int }_{L_2[0,T]}\left|\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right|·\left|{\left(2\pi \right)}^{-\frac{\nu n}{2}}{\left(\frac{2\pi }{\lambda }\right)}^{\frac{\nu n}{2}}\right|d\left|f\right|\left(\overrightarrow{v}\right)\hfill \\ & =& {\lambda }^{-\frac{\nu n}{2}}{\int }_{L_2^{\nu }[0,T]}\left|\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right|d\left|f\right|\left(\overrightarrow{v}\right)\hfill \\ & <& \infty ,\hfill \end{array}$$

(40)

using Lemma 4.  ☐

Theorem 6.

For $z\in {\mathit{C}}_{+}$,

$$\begin{array}{ccc}& & E_{\overrightarrow{x}}^{an{w}_z}\left([D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right)\hfill \\ & =& \underset{n\to \infty }{lim}{z}^{\frac{\nu n}{2}}·E_{\overrightarrow{x}}(exp\{\frac{1-z}{2}\sum _{j=1}^{\nu }\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\left\}[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right).\hfill \end{array}$$

(41)

Proof .

By Lemma 4,

$$\begin{array}{ccc}& & [D,F,z^{-\frac{1}{2}}\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\hfill \\ & =& {\int }_{L_2^{\nu }[0,T]}\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)\hfill \\ & & exp\left\{\sum _{j=1}^{\nu }\right(i{z}^{-\frac{1}{2}}[I,v_j\left(t\right),x_j\left(t\right)]+i[I,v_j\left(t\right),y_j\left(t\right)]\left)\right\}df\left(\overrightarrow{v}\right)\hfill \end{array}$$

(42)

By Lemma 3,

$$\begin{array}{ccc}& & \underset{n\to \infty }{lim}{z}^{\frac{\nu n}{2}}·E_{\overrightarrow{x}}(exp\{\frac{1-z}{2}\sum _{j=1}^{\nu }\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\left\}[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right)\hfill \\ & =& \underset{n\to \infty }{lim}{z}^{\frac{\nu n}{2}}{\int }_{L_2^{\nu }[0,T]}{E}_{\overrightarrow{x}}(exp\{\frac{1-z}{2}\sum _{j=1}^{\nu }\sum _{k=1}^n{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\}\hfill \\ & & ·exp\left\{i\sum _{j=1}^{\nu }[I,v_j\left(t\right),x_j\left(t\right)]\right\})\hfill \\ & & ·\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)·exp\left\{i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\right\}df\left(\overrightarrow{v}\right)\hfill \\ & =& \underset{n\to \infty }{lim}{z}^{\frac{\nu n}{2}}{\int }_{L_2^{\nu }[0,T]}{z}^{-\frac{\nu n}{2}}·exp\{\frac{z-1}{2z}\sum _{j=1}^{\nu }\sum _{k=1}^m{\left[{\int }_0^T{\varphi }_k\left(s\right)v_j\left(s\right)ds\right]}^2-\frac{1}{2}\sum _{j=1}^{\nu }{\int }_0^T{v}_j^2\left(t\right)dt\}\hfill \\ & & ·\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)·exp\left\{i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\right\}df\left(\overrightarrow{v}\right)\hfill \\ & =& \underset{n\to \infty }{lim}{\int }_{L_2^{\nu }[0,T]}\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)·exp\{-\frac{1}{2z}\sum _{j=1}^{\nu }\sum _{k=1}^n{\left[{\int }_0^T{\varphi }_k\left(s\right)v_j\left(s\right)ds\right]}^2\}\hfill \\ & & ·exp\{\frac{1}{2}\sum _{j=1}^{\nu }\sum _{k=1}^n{\left[{\int }_0^T{\varphi }_k\left(s\right)v_j\left(s\right)ds\right]}^2-\frac{1}{2}\sum _{j=1}^{\nu }{\int }_0^T{v}_j^2\left(t\right)dt\}\hfill \\ & & ·exp\left\{i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\right\}df\left(\overrightarrow{v}\right)\hfill \\ & =& {\int }_{L_2^{\nu }[0,T]}\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)·exp\{-\frac{1}{2z}\sum _{j=1}^{\nu }\left|\right|v_j{\left|\right|}_2^2+i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\}df\left(\overrightarrow{v}\right)\hfill \\ & =& {\int }_{L_2^{\nu }[0,T]}\left(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)]\right)·E_{\overrightarrow{x}}(exp\{i{z}^{-\frac{1}{2}}\sum _{j=1}^{\nu }[I,v_j\left(t\right),x_j\left(t\right)]\left\}\right)\hfill \\ & & ·exp\left\{i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\right\}df\left(\overrightarrow{v}\right)\hfill \\ & =& E_{\overrightarrow{x}}\left({\int }_{L_2^{\nu }[0,T]}\right(i\sum _{j=1}^{\nu }[I,v_j\left(t\right),w_j\left(t\right)])\hfill \\ & & ·exp\{i{z}^{-\frac{1}{2}}\sum _{j=1}^{\nu }[I,v_j\left(t\right),x_j\left(t\right)]+i\sum _{j=1}^{\nu }[I,v_j\left(t\right),y_j\left(t\right)]\}df\left(\overrightarrow{v}\right))\hfill \\ & =& E_{\overrightarrow{x}}\left([D,F,z^{-\frac{1}{2}}\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right)\hfill \\ & =& E_{\overrightarrow{x}}^{an{w}_z}\left([D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right).\hfill \end{array}$$

(43)

 ☐

Theorem 7.

For real $\rho >0$,

$$\begin{array}{ccc}& & E_{\overrightarrow{x}}\left([D,F,\rho \overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right)\hfill \\ & =& \underset{n\to \infty }{lim}{\rho }^{-\nu n}·E_{\overrightarrow{x}}(exp\{\frac{{\rho }^2-1}{2{\rho }^2}\sum _{j=1}^{\nu }\sum _{k=1}^m{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\left\}[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right).\hfill \end{array}$$

(44)

Proof. 

For real $\lambda >0$,

$$\begin{array}{ccc}& & E_{\overrightarrow{x}}^{an{w}_{\lambda }}\left([D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right)\hfill \\ & =& E_{\overrightarrow{x}}\left([D,F,{\lambda }^{-\frac{1}{2}}\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right)\hfill \\ & =& \underset{n\to \infty }{lim}{\lambda }^{\frac{\nu n}{2}}·E_{\overrightarrow{x}}(exp\{\frac{1-\lambda }{2}\sum _{j=1}^{\nu }\sum _{k=1}^m{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\left\}[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right).\hfill \end{array}$$

(45)

Taking $\lambda ={\rho }^{-2}$, Equation (44) holds.  ☐

Theorem 8.

$$\begin{array}{ccc}& & E_{\overrightarrow{x}}^{an{f}_q}([D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}])\hfill \\ & =& \underset{n\to \infty }{lim}{{\lambda }_n}^{\frac{\nu n}{2}}·E_{\overrightarrow{x}}(exp\{\frac{1-{\lambda }_n}{2}\sum _{j=1}^{\nu }\sum _{k=1}^m{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\left\}[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right)\hfill \end{array}$$

(46)

whenever $\left\{{\lambda }_n\right\}\to -iq$ through $C_{+}$.

Proof. 

$$\begin{array}{ccc}& & E_{\overrightarrow{x}}^{an{f}_q}([D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}])\hfill \\ & =& \underset{n\to \infty }{lim}{E}_{\overrightarrow{x}}^{an{w}_{{\lambda }_n}}\left([D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right)\hfill \\ & =& \underset{n\to \infty }{lim}{{\lambda }_n}^{\frac{\nu n}{2}}·E_{\overrightarrow{x}}(exp\{\frac{1-{\lambda }_n}{2}\sum _{j=1}^{\nu }\sum _{k=1}^m{[I,{\varphi }_k\left(t\right),x_j\left(t\right)]}^2\left\}[D,F,\overrightarrow{x}+\overrightarrow{y},\overrightarrow{w}]\right)\hfill \end{array}$$

(47)

whenever $\left\{{\lambda }_n\right\}\to -iq$ through $C_{+}$.  ☐

5. Conclusions

We prove very harmonious relationships among the integral transform and function space integrals exploiting the partial derivative on the function space.

Remark 2.

In this paper, we prove new theorems by extending those results in [11,19] to the first variation theory in [1] and to the Integral Transform in [5].

Remark 3.

The author presented this paper in the conference, “The First International Workshop: Constructive Mathematical Analysis” in Selcuk University, Konya, Turkey (2019). Title, abstract and references were introduced in the proceeding (http://constructivemathematicalanalysis.com).