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Article

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

Department of Mathematics, Hanyang University, Seoul 04763, Korea
Submission received: 27 July 2020 / Revised: 25 August 2020 / Accepted: 8 September 2020 / Published: 18 September 2020
(This article belongs to the Section Information Theory, Probability and Statistics)

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.
MSC:
28 C 20

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 ( 0 ) = 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 [ F ( x ) ] = C 0 [ 0 , T ] F ( x ) d m ( x )
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 ρ E M for all ρ > 0 and a scale invariant measurable set N is said to be scale-invariant null provided m ( ρ N ) = 0 for each ρ > 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 G .
Definition 1.
For the definition of the analytic Wiener integral and the analytic Feynman integral, see Definition 1 in [18]: C + = { λ | R e ( λ ) > 0 } and C + = { λ | R e ( λ ) 0 } . For real λ > 0 ,
J F ( λ ) = E x ( F ( λ 1 2 x ) ) .
For each z C + , the analytic Wiener integral is defined by
E x a n w z [ F ( x ) ] = E x ( F ( z 1 2 x ) ) = J F * ( z ) .
Whenever z i q through C + , the analytic Feynman integral is defined by
E x a n f q [ F ( x ) ] = lim z i q E x a n w z [ F ( x ) ] ,
where i 2 = 1 .
Notation 1.
For λ C + and for s a . e . y C 0 [ 0 , T ] , let
( T λ ( F ) ) ( y ) = E x a n w λ [ F ( x + y ) ] .
Definition 2.
For the L 1 -analytic Fourier–Feynmann transform, see Definition 2 in [5]:
( T q ( 1 ) ( F ) ) ( y ) = lim λ i q E x a n w λ [ F ( x + y ) ] = E x a n f q [ F ( x + y ) ] ,
whenever λ i q through C + (if it exists). See [5,9].
Definition 3
(Ref. [1]). The first variation of a Wiener measurable functional F in the direction w C 0 [ 0 , T ] which is defined by the partial derivative as
δ F ( x | w ) = h F ( x + h w ) | h = 0 .
We will denote it by [ D , F , x , w ] .
Remark 1.
For a C + and b R ,
R exp { a u 2 + i b u } d u = π a exp { b 2 4 a } .

3. Results (1). On C 0 [ 0 , T ]

Let
F ( x ) = L 2 [ 0 , T ] exp { i [ I , v ( t ) , x ( t ) ] } d f ( v )
in some Banach algebra S defined on C 0 [ 0 , T ] in [19], where [ I , v ( t ) , x ( t ) ] = 0 T v ( t ) d x ( t ) and assume that L 2 [ 0 , T ] | | v | | 2 d | f | ( v ) < .
Suppose that formulas in this section hold for s a . e . w C 0 [ 0 , T ] and for s a . e . y C 0 [ 0 , T ] .
Lemma 1.
[ D , F , x , w ] = L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) exp { i [ I , v ( t ) , x ( t ) ] } d f ( v ) ,
where [ I , v ( t ) , w ( t ) ] = 0 T v ( t ) d w ( t ) .
Proof .
By Equation (6).
[ D , F , x , w ] = h F ( x + h w ) | h = 0 = h L 2 [ 0 , T ] exp { i h [ I , v ( t ) , w ( t ) ] + i [ I , v ( t ) , x ( t ) ] } d f ( v ) | h = 0 = L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) exp { i [ I , v ( t ) , x ( t ) ] } d f ( v ) .
and
| E w ( L 2 [ 0 , T ] [ I , v ( t ) , w ( t ) ] · exp { i [ I , v ( t ) , x ( t ) ] } d f ( v ) ) | E w ( L 2 [ 0 , T ] | [ I , v ( t ) , w ( t ) ] | d | f | ( v ) ) .
Then
E w ( L 2 [ 0 , T ] | [ I , v ( t ) , w ( t ) ] | d | f | ( v ) ) = L 2 [ 0 , T ] E w ( | [ I , v ( t ) , w ( t ) ] | ) d | f | ( v ) = L 2 [ 0 , T ] [ 1 2 π | v | | 2 2 + | u | exp { u 2 2 | | v | | 2 2 } d u ] d | f | ( v ) = 2 π L 2 [ 0 , T ] | | v | | 2 d | f | ( v ) <
where E w ( F ( w ) ) = C 0 [ 0 , T ] F ( w ) d m ( w ) . So,
L 2 [ 0 , T ] | [ I , v ( t ) , w ( t ) ] | d | f | ( v ) < .
Therefore, [ D , F , x , w ] exists.  ☐
Theorem 1.
[ 1 ] . E x a n w z ( [ D , F , x + y , w ] ) = L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) exp { 1 2 z 0 T v 2 ( s ) d s + i [ I , v ( t ) , y ( t ) ] } d f ( v )
[ 2 ] . E x a n f q ( [ D , F , x + y , w ] ) = L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) exp { i 2 q 0 T v 2 ( s ) d s + i [ I , v ( t ) , y ( t ) ] } d f ( v ) ,
where [ D , F , x + y , w ] = δ F ( x + y | w ) .
Proof. 
[1]. For z C + ,
E x a n w z ( [ D , F , x + y , w ] ) = E x ( [ D , F , z 1 2 x , y , w ] ) = E x ( L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) · exp { i z 1 2 [ I , v ( t ) , x ( t ) ] + i [ I , v ( t ) , y ( t ) ] } d f ( v ) ) = L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) · E x ( exp { i z 1 2 [ I , v ( t ) , x ( t ) ] } ) · exp { i [ I , v ( t ) , y ( t ) ] } d f ( v ) = L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) · exp { 1 2 z | | v | | 2 2 + i [ I , v ( t ) , y ( t ) ] } d f ( v ) .
[2].
E x a n f q ( [ D , F , x + y , w ] ) = lim z i q E x a n w z ( [ D , F , x + y , w ] ) = lim z i q L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) · exp { ( 1 2 z | | v | | 2 2 + i [ I , v ( t ) , y ( t ) ] } d f ( v ) = L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) · exp { i 2 q [ 0 T v 2 ( s ) d s ] + i [ I , v ( t ) , y ( t ) ] } d f ( v ) .
 ☐
Lemma 2.
For λ C + ,
exp { 1 λ 2 k = 1 n [ I , ϕ k ( t ) , x ( t ) ] 2 } · [ D , F , x + y , w ]
is a Wiener integrable function of x C 0 [ 0 , T ] .
Proof. 
E x ( exp { 1 λ 2 k = 1 n [ I , ϕ k ( t ) , x ( t ) ] 2 } [ D , F , x + y , w ] ) = E x ( exp { 1 λ 2 k = 1 n [ I , ϕ k ( t ) , x ( t ) ] 2 } · [ L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) · exp { i [ I , v ( t ) , x ( t ) ] + i [ I , v ( t ) , y ( t ) ] } d f ( v ) ] ) = L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) · E x ( exp { 1 λ 2 k = 1 n [ I , ϕ k ( t ) , x ( t ) ] 2 } · exp { i [ I , v ( t ) , x ( t ) ] + i [ I , v ( t ) , y ( t ) ] } ) d f ( v ) .
and
E x ( exp { 1 λ 2 k = 1 n [ I , ϕ k ( t ) , x ( t ) ] 2 } · exp { i [ I , v ( t ) , x ( t ) ] + i [ I , v ( t ) , y ( t ) ] } ) E x ( exp { 1 λ 2 k = 1 n [ I , ϕ k ( t ) , x ( t ) ] 2 } ) = ( 2 π ) n 2 R n exp { λ 2 k = 1 n u k 2 } d u = ( 2 π ) n 2 · ( 2 π λ ) n 2 = λ n 2 .
Therefore,
| L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) · E x ( exp { 1 λ 2 k = 1 n [ I , ϕ k ( t ) , x ( t ) ] 2 } · exp { i [ I , v ( t ) , x ( t ) ] + i [ I , v ( t ) , y ( t ) ] } ) d f ( v ) | λ n 2 · L 2 [ 0 , T ] | [ I , v ( t ) , w ( t ) ] | d | f | ( v ) .
By the Wiener integration theorem,
E w ( L 2 [ 0 , T ] | [ I , v ( t ) , w ( t ) ] | d | f | ( v ) ) = L 2 [ 0 , T ] [ 1 2 π | | v | | 2 2 · + | u | · exp { u 2 2 | | v | | 2 2 } d u ] d | f | ( v ) = ( 2 π ) 1 2 L 2 [ 0 , T ] | | v | | 2 d | f | ( v ) < .
 ☐
Lemma 3
(Ref. [12]). Let { ϕ j } j = 1 m be an orthonormal set in L 2 [ 0 , T ] . Then for v L 2 [ 0 , T ] and for λ C + ,
E x ( exp { 1 λ 2 k = 1 m [ I , ϕ k ( t ) , x ( t ) ] 2 + i [ I , v ( t ) , x ( t ) ] } ) = λ m 2 · exp { 1 λ 2 λ k = 1 m ( 0 T ϕ k ( s ) v ( s ) d s ) 2 1 2 0 T v 2 ( s ) d s ] } .
Theorem 2.
For z C + ,
E x a n w z ( [ D , F , x + y , w ] ) = lim n z n 2 · E x ( exp { 1 z 2 k = 1 n [ I , ϕ k ( t ) , x ( t ) ] 2 } [ D , F , x + y , w ] )
Proof by Lemma 1.
[ D , F , z 1 2 x + y , w ] = L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) exp { i z 1 2 [ I , v ( t ) , x ( t ) ] + i [ I , v ( t ) , y ( t ) ] } d f ( v ) .
By Lemma 3,
lim n z n 2 · E x ( exp { 1 z 2 k = 1 n [ I , ϕ k ( t ) , x ( t ) ] 2 } [ D , F , x + y , w ] ) = lim n z n 2 L 2 [ 0 , T ] E x ( exp { 1 z 2 k = 1 n [ I , ϕ k ( t ) , x ( t ) ] 2 + i [ I , v ( t ) , x ( t ) ] } ) · ( i [ I , v ( t ) , w ( t ) ] ) · exp { i [ I , v ( t ) , y ( t ) ] } d f ( v ) = lim n z n 2 L 2 [ 0 , T ] z n 2 exp { z 1 2 z k = 1 m 0 T ϕ k ( s ) v ( s ) , d s ] 2 1 2 0 T v 2 ( t ) d t ] } · ( i [ I , v ( t ) , w ( t ) ] ) · exp { i [ I , v ( t ) , y ( t ) ] } d f ( v ) = lim n L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) exp { 1 2 z k = 1 n [ 0 T ϕ k ( s ) v ( s ) d s ] 2 } · exp { 1 2 k = 1 n [ 0 T ϕ k ( s ) v ( s ) d s ] 2 1 2 0 T v 2 ( t ) d t } · exp { i [ I , v ( t ) , y ( t ) ] } d f ( v ) = L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) exp { 1 2 z | | v | | 2 2 + [ I , v ( t ) , y ( t ) ] } d f ( v ) = L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) · E x ( exp { i z 1 2 [ I , v ( t ) , x ( t ) ] } ) · exp { i [ I , v ( t ) , y ( t ) ] } d f ( v ) = E x ( L 2 [ 0 , T ] ( i [ I , v ( t ) , w ( t ) ] ) · exp { i z 1 2 [ I , v ( t ) , x ( t ) ] + i [ I , v ( t ) , y ( t ) ] } d f ( v ) ) = E x ( [ D , F , z 1 2 x + y , w ] ) = E x a n w z ( [ D , F , x + y , w ] ) .
 ☐
Theorem 3.
For real ρ > 0 ,
E x ( [ D , F , ρ x + y , w ] ) = lim n ρ n · E x ( exp { ρ 2 1 2 ρ 2 k = 1 m [ I , ϕ k ( t ) , x ( t ) ] 2 } [ D , F , x + y , w ] ) .
Proof. 
For real λ > 0 ,
E x a n w λ ( [ D , F , x + y , w ] ) E x ( [ D , F , λ 1 2 x + y , w ] ) = lim n λ n 2 · E x ( exp { 1 λ 2 k = 1 m [ I , ϕ k ( t ) , x ( t ) ] 2 } [ D , F , x + y , w ] ) .
Taking λ = ρ 2 , we have the result.  ☐
Theorem 4.
E x a n f q ( [ D , F , x + y , w ] ) = lim n λ n n 2 · E x ( exp { 1 λ n 2 k = 1 m [ I , ϕ k ( t ) , x ( t ) ] 2 } [ D , F , x + y , w ] ) ,
whenever { λ n } i q through C + .
Proof .
by Theorem 2,
E x a n f q ( [ D , F , x + y , w ] ) = lim n E x a n w λ n ( [ D , F , x + y , w ] ) = lim n λ n n 2 · E x ( exp { 1 λ n 2 k = 1 m [ I , ϕ k ( t ) , x ( t ) ] 2 } [ D , F , x + y , w ] ) .
 ☐

4. Results (2). on C 0 ν [ 0 , T ]

In this section, we expand the result about the function:
F ( x ) = L 2 ν [ 0 , T ] exp { i j = 1 ν [ I , v j ( t ) , x j ( t ) ] } d f ( v )
in some Banach algebra S defined on C 0 ν [ 0 , T ] in [14].
Let w = ( w 1 , , w ν ) , where w j C 0 [ 0 , T ] is absolutely continuous on [ 0 , T ] and w j ( t ) L 2 [ 0 , T ] for 1 j ν . Suppose also that M = M a x 1 j ν | | w j | | 2 < and we assume that L 2 ν [ 0 , T ] j = 1 ν | | v j | | 2 d | f | ( v ) < .
Suppose that formulas in this section hold s a . e . w C 0 ν [ 0 , T ] and for s a . y C 0 ν [ 0 , T ] .
Lemma 4.
[ D , F , x , w ] = L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) exp { i j = 1 ν [ I , v j ( t ) , x j ( t ) ] } d f ( v ) .
Proof .
By Equation (6).
[ D , F , x , w ] = h F ( x + h w ) | h = 0 = h L 2 ν [ 0 , T ] exp { i h j = 1 ν [ I , v j ( t ) , w j ( t ) ] + i j = 1 ν [ I , v j ( t ) , x j ( t ) ] } d f ( v ) | h = 0 = L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) exp { i j = 1 ν [ I , v j ( t ) , x j ( t ) ] } d f ( v ) .
We know that the Paley–Wiener–Zygmund integral equals to the Riemann–Stieltzes integral
0 T f ( t ) d g ( t ) = 0 T f ( t ) g ( t ) d t ,
if g is absolutely continuous in [ 0 , T ] with g ( t ) L 2 [ 0 , T ] .
For 1 j ν , 0 T v j ( t ) d w j ( t ) = 0 T v j ( t ) w j ( t ) d t . Therefore
| L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) exp { i j = 1 ν [ I , v j ( t ) , x j ( t ) ] } d f ( v ) | L 2 ν [ 0 , T ] | j = 1 ν 0 T v j ( t ) w j ( t ) d t | d | f | ( v ) L 2 ν [ 0 , T ] j = 1 ν | | v j | | 2 · | | w j | | 2 d | f | ( v ) L 2 ν [ 0 , T ] j = 1 ν | | v j | | 2 · [ M a x 1 j ν | | w j | | 2 ] d | f | ( v ) = M · L 2 ν [ 0 , T ] j = 1 ν | | v j | | 2 d | f | ( v ) < ,
where M = M a x 1 j ν | | w j | | 2 < .  ☐
Theorem 5.
( 1 ) . E x a n w z ( [ D , F , x + y , w ] ) = L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) exp { 1 2 z j = 1 ν 0 T v j 2 ( s ) d s + i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v ) .
( 2 ) . E x a n f q ( [ D , F , x + y , w ] ) = L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) exp { i 2 q j = 1 ν 0 T v j 2 ( s ) d s + i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v ) .
Proof .
( 1 ) . For z C + ,
E x a n w z ( [ D , F , x + y , w ] ) = E x ( [ D , F , z 1 2 x + y , w ] ) = E x ( L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) · exp { i z 1 2 [ I , v j ( t ) , x j ( t ) ] + i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v ) ) = L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) · [ E x ( exp { i z 1 2 j = 1 ν [ I , v j ( t ) , x j ( t ) ] } ) · exp { i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v ) = L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) · exp { 1 2 z j = 1 ν | | v j | | 2 2 } · exp { i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v ) = L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) exp { 1 2 z j = 1 ν | | v j | | 2 2 + i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v )
( 2 ) .
E x a n f q ( [ D , F , x + y , w ] ) = lim z i q E x a n w z ( [ D , F , x + y , w ] ) = lim z i q L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) · exp { 1 2 z j = 1 ν | | v j | | 2 2 + i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v ) = L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) · exp { i 2 q j = 1 ν 0 T v j 2 ( s ) d s + i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v )
 ☐
Lemma 5.
For λ C + ,
exp { 1 λ 2 j = 1 ν k = 1 n [ I , ϕ k ( t ) , x j ( t ) ] 2 } [ D , F , x + y , w ]
is a Wiener-integrable function of x C 0 ν [ 0 , T ] .
Proof. 
First we have
E x ( exp { 1 λ 2 j = 1 ν k = 1 n [ I , ϕ k ( t ) , x j ( t ) ] 2 } [ D , F , x + y , w ] ) = E x ( exp { 1 λ 2 k = 1 n ( j = 1 ν [ I , ϕ k ( t ) , x j ( t ) ] 2 } · [ L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) · exp { i j = 1 ν [ I , v j ( t ) , x ( t ) ] + i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v ) ) = L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) · E x ( exp { 1 λ 2 j = 1 ν k = 1 n [ I , ϕ k ( t ) , x j ( t ) ] 2 } · exp { i j = 1 ν [ I , v j ( t ) , x j ( t ) ] + i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } ) d f ( v ) .
Therefore we have
| E x ( exp { 1 λ 2 j = 1 ν k = 1 n [ I , ϕ k ( t ) , x j ( t ) ] 2 } [ D , F , x + y , w ] ) | L 2 ν [ 0 , T ] | j = 1 ν [ I , v j ( t ) , w j ( t ) ] | · E x ( | exp { 1 λ 2 j = 1 ν k = 1 n [ I , ϕ k ( t ) , x j ( t ) ] 2 } | ) d | f | ( v ) L 2 ν [ 0 , T ] | j = 1 ν [ I , v j ( t ) , w j ( t ) ] | · | ( 2 π ) n 2 R ν n exp { λ 2 j = 1 ν k = 1 n u j , k 2 } d u | d | f | ( v ) = L 2 [ 0 , T ] | j = 1 ν [ I , v j ( t ) , w j ( t ) ] | · | ( 2 π ) ν n 2 ( 2 π λ ) ν n 2 | d | f | ( v ) = λ ν n 2 L 2 ν [ 0 , T ] | j = 1 ν [ I , v j ( t ) , w j ( t ) ] | d | f | ( v ) < ,
using Lemma 4.  ☐
Theorem 6.
For z C + ,
E x a n w z ( [ D , F , x + y , w ] ) = lim n z ν n 2 · E x ( exp { 1 z 2 j = 1 ν k = 1 n [ I , ϕ k ( t ) , x j ( t ) ] 2 } [ D , F , x + y , w ] ) .
Proof .
By Lemma 4,
[ D , F , z 1 2 x + y , w ] = L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) exp { j = 1 ν ( i z 1 2 [ I , v j ( t ) , x j ( t ) ] + i [ I , v j ( t ) , y j ( t ) ] ) } d f ( v )
By Lemma 3,
lim n z ν n 2 · E x ( exp { 1 z 2 j = 1 ν k = 1 n [ I , ϕ k ( t ) , x j ( t ) ] 2 } [ D , F , x + y , w ] ) = lim n z ν n 2 L 2 ν [ 0 , T ] E x ( exp { 1 z 2 j = 1 ν k = 1 n [ I , ϕ k ( t ) , x j ( t ) ] 2 } · exp { i j = 1 ν [ I , v j ( t ) , x j ( t ) ] } ) · ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) · exp { i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v ) = lim n z ν n 2 L 2 ν [ 0 , T ] z ν n 2 · exp { z 1 2 z j = 1 ν k = 1 m [ 0 T ϕ k ( s ) v j ( s ) d s ] 2 1 2 j = 1 ν 0 T v j 2 ( t ) d t } · ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) · exp { i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v ) = lim n L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) · exp { 1 2 z j = 1 ν k = 1 n [ 0 T ϕ k ( s ) v j ( s ) d s ] 2 } · exp { 1 2 j = 1 ν k = 1 n [ 0 T ϕ k ( s ) v j ( s ) d s ] 2 1 2 j = 1 ν 0 T v j 2 ( t ) d t } · exp { i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v ) = L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) · exp { 1 2 z j = 1 ν | | v j | | 2 2 + i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v ) = L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) · E x ( exp { i z 1 2 j = 1 ν [ I , v j ( t ) , x j ( t ) ] } ) · exp { i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v ) = E x ( L 2 ν [ 0 , T ] ( i j = 1 ν [ I , v j ( t ) , w j ( t ) ] ) · exp { i z 1 2 j = 1 ν [ I , v j ( t ) , x j ( t ) ] + i j = 1 ν [ I , v j ( t ) , y j ( t ) ] } d f ( v ) ) = E x ( [ D , F , z 1 2 x + y , w ] ) = E x a n w z ( [ D , F , x + y , w ] ) .
 ☐
Theorem 7.
For real ρ > 0 ,
E x ( [ D , F , ρ x + y , w ] ) = lim n ρ ν n · E x ( exp { ρ 2 1 2 ρ 2 j = 1 ν k = 1 m [ I , ϕ k ( t ) , x j ( t ) ] 2 } [ D , F , x + y , w ] ) .
Proof. 
For real λ > 0 ,
E x a n w λ ( [ D , F , x + y , w ] ) = E x ( [ D , F , λ 1 2 x + y , w ] ) = lim n λ ν n 2 · E x ( exp { 1 λ 2 j = 1 ν k = 1 m [ I , ϕ k ( t ) , x j ( t ) ] 2 } [ D , F , x + y , w ] ) .
Taking λ = ρ 2 , Equation (44) holds.  ☐
Theorem 8.
E x a n f q ( [ D , F , x + y , w ] ) = lim n λ n ν n 2 · E x ( exp { 1 λ n 2 j = 1 ν k = 1 m [ I , ϕ k ( t ) , x j ( t ) ] 2 } [ D , F , x + y , w ] )
whenever { λ n } i q through C + .
Proof. 
E x a n f q ( [ D , F , x + y , w ] ) = lim n E x a n w λ n ( [ D , F , x + y , w ] ) = lim n λ n ν n 2 · E x ( exp { 1 λ n 2 j = 1 ν k = 1 m [ I , ϕ k ( t ) , x j ( t ) ] 2 } [ D , F , x + y , w ] )
whenever { λ n } i q 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).

Funding

Fund of this paper was supported by NRF-2017R1A6A3A11030667.

Conflicts of Interest

The author declares no conflict of interest.

References

  1. Cameron, R.H. The first variation of an indefinite Wiener integral. Proc. Am. Soc. 1951, 2, 914–924. [Google Scholar] [CrossRef]
  2. Cameron, R.H. The translation pathology of Wiener space. Duke Math. J. 1954, 21, 623–628. [Google Scholar] [CrossRef]
  3. Cameron, R.H.; Martin, W.T. On transformations of Wiener integrals under translations. Ann. Math. 1944, 45, 386–396. [Google Scholar] [CrossRef]
  4. Cameron, R.H.; Martin, W.T. Transformations for Wiener integrals under a general class of linear transformations. Trans. Am. Math. Soc. 1945, 58, 184–219. [Google Scholar] [CrossRef] [Green Version]
  5. Cameron, R.H.; Storvick, D.A. An L2-analytic Fourier Feynman transforms. Mich. Math. J. 1976, 23, 1–30. [Google Scholar]
  6. Huffman, T.; Park, C.; Skoug, D. Analytic Fourier Feynman transforms and convolution. Trans. Am. Math. Soc. 1995, 347, 661–673. [Google Scholar] [CrossRef]
  7. Huffman, T.; Park, C.; Skoug, D. Convolution and Fourier Feynman transforms of functions involving multiple integrals. Mich. Math. J. 1996, 43, 247–261. [Google Scholar]
  8. Huffman, T.; Park, C.; Skoug, D. Convolution and Fourier Feynman transforms. Rocky Mt. J. Math. 1997, 27, 827–841. [Google Scholar] [CrossRef]
  9. Johnson, G.W.; Skoug, D.L. An Lp-analytic Fourier Feynman transforms. Mich. Math. J. 1979, 26, 103–127. [Google Scholar]
  10. Cameron, R.H.; Martin, W.T. The behavior of measure and measurability under change of scale in Wiener space. Bull. Am. Math. Soc. 1947, 53, 130–137. [Google Scholar] [CrossRef] [Green Version]
  11. Cameron, R.H.; Storvick, D.A. Change of scale formulas for Wiener integral. Suppl. Rendicoti Circ. Mat. Palermo Ser. II—Numero 1987, 17, 105–115. [Google Scholar]
  12. Cameron, R.H.; Storvick, D.A. Relationships between the Wiener integral and the analytic Feynman integral. Suppl. Rendicoti Circ. Mat. Palermo Ser. II—Numero 1988, 17, 117–133. [Google Scholar]
  13. Kim, Y.S. Relationships between Fourier Feynman transforms and Wiener integrals on abstract Wiener spaces. Integral Transform. Spec. Funct. 2001, 12, 323–332. [Google Scholar] [CrossRef]
  14. Kim, Y.S. Behavior of a scale factor for Wiener integrals and a Fourier Stieltjes transform on the Wiener space. Appl. Math. 2018, 9, 488–495. [Google Scholar] [CrossRef] [Green Version]
  15. Kim, Y.S. Relationships between Fourier Feynman transforms and Wiener integrals on abstract Wiener spaces II. Integral Transform. Spec. Funct. 2005, 16, 57–64. [Google Scholar] [CrossRef]
  16. Kim, Y.S. Behavior of the first variation of a measure on the Fourier Feynman Transform and Convolution. Numer. Funct. Anal. Optim. 2016, 37, 699–718. [Google Scholar] [CrossRef]
  17. Kim, Y.S. The behavior of the first variation under the Fourier Feynman transform on abstract Wiener spaces. J. Fourier Anal. Appl. 2006, 12, 233–242. [Google Scholar] [CrossRef]
  18. Kim, Y.S. Fourier Feynman Transform and analytic Feynman integrals and convolutions of a Fourier transform μ of a measure on Wiener spaces. Houst. J. Math. 2010, 36, 1139–1158. [Google Scholar]
  19. Cameron, R.H.; Storvick, D.A. Some Banach algebras of analytic Feynman integrable functionals, an analytic functions. Lect. Notes Math. 1980, 798, 18–27. [Google Scholar]

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Young Sik, K. Partial Derivative Approach to the Integral Transform for the Function Space in the Banach Algebra. Entropy 2020, 22, 1047. https://0-doi-org.brum.beds.ac.uk/10.3390/e22091047

AMA Style

Young Sik K. Partial Derivative Approach to the Integral Transform for the Function Space in the Banach Algebra. Entropy. 2020; 22(9):1047. https://0-doi-org.brum.beds.ac.uk/10.3390/e22091047

Chicago/Turabian Style

Young Sik, Kim. 2020. "Partial Derivative Approach to the Integral Transform for the Function Space in the Banach Algebra" Entropy 22, no. 9: 1047. https://0-doi-org.brum.beds.ac.uk/10.3390/e22091047

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