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

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

Kim Young Sik

Department of Mathematics, Hanyang University, Seoul 04763, Korea; [email protected]

Received: 27 July 2020; Accepted: 8 September 2020; Published: 18 September 2020

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.

Keywords: function space; integral transform

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 [2–4]. Integral transforms for the function space were expanded upon in [5–9]. A change of scale formula and a scale factor for the Wiener integral were expanded in [10–12] 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 C0[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 C0[0, T] and let m denote the Wiener measure. Then (C0[0, T], M, m) is a complete measure space and Z Ex[F(x)] = F(x) dm(x) C0[0,T] is called the Wiener integral of a function F deﬁned on the function space C0[0, T]. A subset E of C0[0, T] is said to be scale-invariant measurable provided ρE ∈ M for all ρ > 0 and a scale invariant measurable set N is said to be scale-invariant null provided m(ρN) = 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|Re(λ) > 0} and C+ = {λ|Re(λ) ≥ 0}. For real λ > 0, − 1 JF(λ) = Ex F(λ 2 x) . (1)

Entropy 2020, 22, 1047; doi:10.3390/e22091047 www.mdpi.com/journal/entropy Entropy 2020, 22, 1047 2 of 13

For each z ∈ C+, the analytic Wiener integral is deﬁned by

1 anwz − ∗ Ex [F(x)] = Ex F(z 2 x) = JF(z). (2)

Whenever z → −i q through C+, the analytic Feynman integral is deﬁned by

an fq anwz Ex [F(x)] = lim Ex [F(x)], (3) z→−iq where i2 = −1.

Notation 1. For λ ∈ C+ and for s − a.e.y ∈ C0[0, T], let

anwλ ( Tλ(F))(y) = Ex [F( x + y )]. (4)

Deﬁnition 2. For the L1-analytic Fourier–Feynmann transform, see Deﬁnition 2 in [5]:

(1) anwλ an fq ( Tq (F))(y) = lim Ex [F( x + y )] = Ex [F( x + y )] , (5) λ→−iq 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 ∈ C0[0, T] which is defined by the partial derivative as

∂ δF(x|w) = F(x + hw)| . (6) ∂h h=0

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

Remark 1. For a ∈ C+ and b ∈ R,

Z r π b2 exp{−au2 + ibu} du = exp{− } . (7) R a 4a

3. Results (1). On C0[0, T] Let Z F(x) = exp i [ I, v(t), x(t)] d f (v) (8) L2[0,T]

R T in some Banach algebra S deﬁned on C0[0, T] in [19], where [I, v(t), x(t)] = 0 v(t)dx(t) and assume that R ||v|| d| f |(v) < ∞. L2[0,T] 2 Suppose that formulas in this section hold for s − a.e.w ∈ C0[0, T] and for s − a.e.y ∈ C0[0, T].

Lemma 1. Z [D, F, x, w] = i [I, v(t), w(t)] exp i [I, v(t), x(t)] d f (v), (9) L2[0,T]

R T where [I, v(t), w(t)] = 0 v(t)dw(t). Entropy 2020, 22, 1047 3 of 13

Proof. By Equation (6).

[D, F, x, w] ∂ = F( x + hw )| ∂h h=0 ∂ Z = exp i h [I, v(t), w(t)] + i [I, v(t), x(t)] d f (v) |h=0 ∂h L2[0,T] Z = i [I, v(t), w(t)] exp i [I, v(t), x(t)] d f (v) . (10) L2[0,T] and Z Ew [I, v(t), w(t)] · exp i [I, v(t), x(t)] d f (v) L2[0,T] Z ≤ Ew [I, v(t), w(t)] d| f |(v) . (11) L2[0,T]

Then Z Ew [I, v(t), w(t)] d| f |(v) L2[0,T] Z = Ew [I, v(t), w(t)] d| f |(v) L2[0,T] Z 1 Z +∞ u2 = | | − | |( ) q u exp 2 du d f v L2[0,T] 2 −∞ 2||v|| 2π |v||2 2 r 2 Z = ||v||2 d| f |(v) < ∞ (12) π L2[0,T] R where Ew F(w) = F(w) dm(w). So, C0[0,T]

Z [I, v(t), w(t)] d| f |(v) < ∞. L2[0,T]

Therefore, [D, F, x, w] exists.

Theorem 1. anwz [1]. Ex [D, F, x + y, w]

Z 1 Z T = i [I, v(t), w(t)] exp − v2(s)ds + i [I, v(t), y(t)] d f (v) (13) L2[0,T] 2z 0

an fq [2]. Ex [D, F, x + y, w]

Z i Z T = i [I, v(t), w(t)] exp − v2(s)ds + i [I, v(t), y(t)] d f (v) , (14) L2[0,T] 2q 0 where [D, F, x + y, w] = δF(x + y|w). Entropy 2020, 22, 1047 4 of 13

Proof. [1]. For z ∈ C+, anwz Ex [D, F, x + y, w] − 1 = Ex [D, F, z 2 x , y, w ]

Z = Ex i [I, v(t), w(t)] L2[0,T] − 1 · exp iz 2 [I, v(t), x(t)] + i [I, v(t), y(t)] d f (v)

Z = i [I, v(t), w(t)] L2[0,T] − 1 ·Ex exp iz 2 [I, v(t), x(t)] · exp i [I, v(t), y(t)] d f (v)

Z 1 2 = i [I, v(t), w(t)] · exp − ||v||2 + i [I, v(t), y(t)] d f (v) . (15) L2[0,T] 2z

[2]. an fq Ex [D, F, x + y, w] anwz = lim Ex [D, F, x + y, w ] z→−iq Z 1 2 = lim i [I, v(t), w(t)] · exp ( − ||v||2 + i [I, v(t), y(t)] d f (v) z→−iq L2[0,T] 2z Z i Z T = i [I, v(t), w(t)] · exp − [ v2(s)ds] + i[I, v(t), y(t)] d f (v) . (16) L2[0,T] 2q 0

Lemma 2. For λ ∈ C+,

n 1 − λ 2 exp ∑ [I, φk(t), x(t)] · [D, F, x + y, w] (17) 2 k=1 is a Wiener integrable function of x ∈ C0[0, T].

Proof.

n 1 − λ 2 Ex exp ∑ [I, φk(t), x(t)] [D, F, x + y, w] 2 k=1 n 1 − λ 2 = Ex exp ∑ [I, φk(t), x(t)] 2 k=1 Z · i [I, v(t), w(t)] · exp i [I, v(t), x(t)] + i [I, v(t), y(t)] d f (v) L2[0,T] Z n 1 − λ 2 = i [I, v(t), w(t)] · Ex exp ∑ [I, φk(t), x(t)] L2[0,T] 2 k=1 · exp i [I, v(t), x(t)] + i [I, v(t), y(t)] d f (v) . (18) Entropy 2020, 22, 1047 5 of 13 and

n 1 − λ 2 Ex exp ∑ [I, φk(t), x(t)] · exp i [I, v(t), x(t)] + i [I, v(t), y(t)] 2 k=1 n 1 − λ 2 ≤ Ex exp ∑ [I, φk(t), x(t)] 2 k=1 Z n − n λ 2 = (2π) 2 exp − u d~u n ∑ k R 2 k=1 − n 2π n = (2π) 2 · ( ) 2 λ − n = λ 2 . (19)

Therefore,

Z n 1 − λ 2 i [I, v(t), w(t)] · Ex exp ∑ [I, φk(t), x(t)] L2[0,T] 2 k=1

· exp i [I, v(t), x(t)] + i [I, v(t), y(t)] d f (v)

Z − n ≤ λ 2 · [I, v(t), w(t)] d| f |(v) . (20) L2[0,T]

By the Wiener integration theorem, Z Ew [I, v(t), w(t)] d| f |(v) L2[0,T] Z 1 Z +∞ u2 = · | | · − | |( ) q u exp 2 du d f v L2[0,T] 2 −∞ 2||v|| 2 π||v||2 2 2 1 Z = ( ) 2 ||v||2 d| f |(v) < ∞ . (21) π L2[0,T]

m Lemma 3 (Ref. [12]). Let {φj}j=1 be an orthonormal set in L2[0, T]. Then for v ∈ L2[0, T] and for λ ∈ C+,

m 1 − λ 2 Ex exp ∑ [I, φk(t), x(t)] + i [I, v(t), x(t)] 2 k=1 m Z T Z T − m 1 − λ 2 1 2 = λ 2 · exp − ∑ ( φk(s)v(s)ds ) − v (s)ds ] . (22) 2λ k=1 0 2 0

Theorem 2. For z ∈ C+, anwz Ex [D, F, x + y, w]

n n 1 − z 2 = lim z 2 · Ex exp [I, φk(t) , x(t)] [D, F, x + y, w] (23) n→∞ ∑ 2 k=1 Entropy 2020, 22, 1047 6 of 13

Proof by Lemma 1.

− 1 [D, F, z 2 x + y, w] Z − 1 = i [I, v(t), w(t)] exp i z 2 [I, v(t), x(t)] + i [I, v(t), y(t)] d f (v) . (24) L2[0,T]

By Lemma 3,

n n 1 − z 2 lim z 2 · Ex exp [I, φk(t), x(t)] [D, F, x + y, w] n→∞ ∑ 2 k=1 Z n n 1 − z 2 = lim z 2 Ex exp [I, φk(t), x(t)] + i [I, v(t), x(t)] n→∞ ∑ L2[0,T] 2 k=1 · i [I, v(t), w(t)] · exp i [I, v(t), y(t)] d f (v)

Z m Z T Z T n − n z − 1 2 1 2 = lim z 2 z 2 exp φk(s) v(s), ds ] − v (t)dt] n→∞ ∑ L2[0,T] 2 z k=1 0 2 0 · i [I, v(t), w(t)] · exp i [I, v(t), y(t)] d f (v)

Z n Z T 1 2 = lim i [ I, v(t), w(t)] exp − [ φk(s) v(s) ds ] n→∞ ∑ L2[0,T] 2 z k=1 0 n Z T Z T 1 2 1 2 · exp ∑ [ φk(s) v(s) ds ] − v (t) dt 2 k=1 0 2 0 · exp i [I, v(t), y(t)] d f (v)

Z 1 2 = i [ I, v(t), w(t)] exp − ||v||2 + [I, v(t), y(t)] d f (v) L2[0,T] 2z Z − 1 = i [I, v(t), w(t)] · Ex exp iz 2 [I, v(t), x(t)] L2[0,T] · exp i [I, v(t), y(t)] d f (v)

Z = Ex i [I, v(t), w(t)] L2[0,T] − 1 · exp iz 2 [I, v(t), x(t)] + i [I, v(t), y(t)] d f (v) − 1 = Ex [D, F, z 2 x + y, w] anwz = Ex [D, F, x + y, w] . (25)

Theorem 3. For real ρ > 0, Ex [D, F, ρ x + y, w]

2 m −n ρ − 1 2 = lim ρ · Ex exp [ I, φk(t), x(t)] [D, F, x + y, w] . (26) n→∞ 2 ∑ 2ρ k=1 Entropy 2020, 22, 1047 7 of 13

Proof. For real λ > 0, anwλ Ex [D, F, x + y, w] − 1 ≡ Ex [D, F, λ 2 x + y, w]

m n 1 − λ 2 = lim λ 2 · Ex exp [I, φk(t), x(t)] [ D, F, x + y, w ] . (27) n→∞ ∑ 2 k=1

Taking λ = ρ−2, we have the result.

Theorem 4. an fq Ex [D, F, x + y, w]

m n 1 − λn 2 = lim λn 2 · Ex exp [I, φk(t), x(t)] [ D, F, x + y, w] , (28) n→∞ ∑ 2 k=1 whenever {λn} → − iq through C+.

Proof. by Theorem 2, an fq Ex [D, F, x + y, w] anwλn = lim Ex [D, F, x + y, w] n→∞ m n 1 − λn 2 = lim λn 2 · Ex exp [I, φk(t), x(t)] [ D, F, x + y, w] . (29) n→∞ ∑ 2 k=1

ν 4. Results (2). on C0 [0, T] In this section, we expand the result about the function :

Z ν F(~x) = exp i [ I, vj(t), xj(t)] d f (~v) (30) ν[ ] ∑ L2 0,T j=1 in some Banach algebra S0 deﬁned on Cν[0, T] in [14]. 0 0 Let w~ = (w1, ··· , wν), where wj ∈ C0[0, T] is absolutely continuous on [0, T] and wj(t) ∈ L2[0, T] 0 for 1 ≤ j ≤ ν. Suppose also that M = Max1≤j≤ν||wj||2 < ∞ and we assume that R ν ν ∑j= ||vj||2 d| f |(~v) < ∞. L2[0,T] 1 ν ν Suppose that formulas in this section hold s − a.e.w~ ∈ C0 [0, T] and for s − a.~y ∈ C0 [0, T].

Lemma 4.

Z ν ν [D, F,~x, w~ ] = i [ I, vj(t), wj(t)] exp i [ I, vj(t), xj(t)] d f (~v). (31) ν[ ] ∑ ∑ L2 0,T j=1 j=1 Entropy 2020, 22, 1047 8 of 13

Proof. By Equation (6).

[D, F,~x, w~ ] ∂ = F(~x + hw~ )| ∂h h=0 ∂ Z ν ν = exp i h [ I, vj(t), wj(t) ] + i [ I, vj(t), xj(t)] d f (~v) |h=0 ∂h ν[ ] ∑ ∑ L2 0,T j=1 j=1 Z ν ν = i [ I, vj(t), wj(t)] exp i [ I, vj(t), xj(t)] d f (~v) . (32) ν[ ] ∑ ∑ L2 0,T j=1 j=1

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

Z T Z T f (t) dg(t) = f (t) g0(t) dt , 0 0

0 if g is absolutely continuous in [0, T] with g (t) ∈ L2[0, T]. R T R T 0 For 1 ≤ j ≤ ν, 0 vj(t)dwj(t) = 0 vj(t)wj(t) dt. Therefore

ν ν Z i [ I, vj(t), wj(t)] exp i [ I, vj(t), xj(t)] d f (~v) ν[ ] ∑ ∑ L2 0,T j=1 j=1 ν Z Z T 0 ≤ vj(t) wj(t) dt d| f |(~v) ν[ ] ∑ L2 0,T j=1 0

Z ν 0 ≤ ||vj||2 · ||wj||2 d| f |(~v) ν[ ] ∑ L2 0,T j=1

Z ν 0 ≤ ||vj||2 · [ Max1≤j≤ν||wj||2 ] d| f |(~v) ν[ ] ∑ L2 0,T j=1 Z ν = M · ||vj||2 d| f |(~v) ν[ ] ∑ L2 0,T j=1 < ∞ , (33)

0 where M = Max1≤j≤ν||wj||2 < ∞.

Theorem 5. anwz ~ ~ ~ (1). E~x [D, F, x + y, w ]

Z ν ν Z T ν (34) 1 2 = i [I, vj(t), wj(t)] exp − vj (s)ds + i [I, vj(t), yj(t)] d f (~v). ν[ ] ∑ z ∑ ∑ L2 0,T j=1 2 j=1 0 j=1

an fq ~ ~ ~ (2). E~x [D, F, x + y, w]

Z ν ν Z T ν (35) i 2 = i [ I, vj(t), wj(t)] exp − vj (s)ds + i [ I, vj(t), yj(t)] d f (~v). ν[ ] ∑ q ∑ ∑ L2 0,T j=1 2 j=1 0 j=1 Entropy 2020, 22, 1047 9 of 13

Proof. (1). For z ∈ C+, anwz ~ ~ ~ E~x [D, F, x + y, w ] − 1 = E~x [D, F, z 2 ~x +~y, w~ ]

Z ν = E~x i [ I, vj(t), wj(t)] ν[ ] ∑ L2 0,T j=1 ν − 1 · exp iz 2 [ I, vj(t), xj(t) ] + i ∑ [ I, vj(t), yj(t)] d f (~v) j=1 Z ν ν − 1 = i [ I, vj(t), wj(t)] · E~x exp{iz 2 [ I, vj(t), xj(t)] ν[ ] ∑ ∑ L2 0,T j=1 j=1 ν · exp i ∑ [ I, vj(t), yj(t)] d f (~v) j=1 Z ν ν 1 2 = i [I, vj(t), wj(t)] · exp − ||vj||2 ν[ ] ∑ z ∑ L2 0,T j=1 2 j=1 ν · exp i ∑ [ I, vj(t), yj(t)] d f (~v) j=1 Z ν ν ν 1 2 = i [I, vj(t), wj(t)] exp − ||vj||2 + i [I, vj(t), yj(t)] d f (~v) (36) ν[ ] ∑ z ∑ ∑ L2 0,T j=1 2 j=1 j=1

(2). an fq ~ ~ ~ E~x [D, F, x + y, w] = lim Eanwz [D, F,~x +~y, w~ ] z→−iq ~x Z ν ν ν 1 2 = lim i [I, vj(t), wj(t)] · exp − ||vj||2 + i [I, vj(t), yj(t)] d f (~v) z→−iq ν[ ] ∑ z ∑ ∑ L2 0,T j=1 2 j=1 j=1 Z ν ν Z T ν i 2 = i [I, vj(t), wj(t)] · exp − vj (s)ds + i [I, vj(t), yj(t)] d f (~v) ν[ ] ∑ q ∑ ∑ L2 0,T j=1 2 j=1 0 j=1 (37)

Lemma 5. For λ ∈ C+,

ν n 1 − λ 2 exp ∑ ∑ [I, φk(t), xj(t)] [D, F,~x +~y, w~ ] (38) 2 j=1 k=1

ν is a Wiener-integrable function of ~x ∈ C0 [0, T]. Entropy 2020, 22, 1047 10 of 13

Proof. First we have

ν n 1 − λ 2 E~x exp ∑ ∑ [ I, φk(t), xj(t)] [D, F,~x +~y, w~ ] 2 j=1 k=1 n ν 1 − λ 2 = E~x exp ∑ (∑[I, φk(t), xj(t)] 2 k=1 j=1 Z ν · i [I, vj(t) , wj(t)] ν[ ] ∑ L2 0,T j=1 ν ν · exp i ∑ [I, vj(t), x(t) ] + i ∑ [I, vj(t), yj(t)] d f (~v) j=1 j=1 Z ν ν n 1 − λ 2 = i [ I, vj(t) , wj(t)] · E~x exp [I, φk(t), xj(t)] ν[ ] ∑ 2 ∑ ∑ L2 0,T j=1 j=1 k=1 ν ν · exp i ∑ [I, vj(t), xj(t) ] + i ∑ [I, vj(t), yj(t)] d f (~v) . (39) j=1 j=1

Therefore we have

ν n 1 − λ 2 E~x exp ∑ ∑ [ I, φk(t), xj(t)] [D, F,~x +~y, w~ ] 2 j=1 k=1 Z ν ν n 1 − λ 2 ≤ [I, vj(t), wj(t)] · E~x exp [I, φk(t), xj(t)] d| f |(~v) ν[ ] ∑ 2 ∑ ∑ L2 0,T j=1 j=1 k=1 ν ν n Z n Z λ − 2 2 ≤ [I, vj(t), wj(t)] · (2π) exp − uj,k d~u d| f |(~v) ν[ ] ∑ νn 2 ∑ ∑ L2 0,T j=1 R j=1 k=1 Z ν − νn 2π νn = ∑ [I, vj(t), wj(t)] · (2π) 2 ( ) 2 d| f |(~v) L2[0,T] j=1 λ Z ν − νn = λ 2 [I, vj(t), wj(t)] d| f |(~v) ν[ ] ∑ L2 0,T j=1 < ∞ , (40) using Lemma 4.

Theorem 6. For z ∈ C+, anwz ~ ~ ~ E~x [D, F, x + y, w ] ν n νn 1 − z 2 = lim z 2 · E~x exp [I, φk(t), xj(t)] [D, F,~x +~y, w~ ] . (41) n→∞ ∑ ∑ 2 j=1 k=1

Proof. By Lemma 4,

− 1 [D, F, z 2 ~x +~y, w~ ] Z ν = i [I, vj(t), wj(t)] ν[ ] ∑ L2 0,T j=1 ν − 1 exp ∑ iz 2 [I, vj(t), xj(t)] + i[I, vj(t), yj(t)] d f (~v) (42) j=1 Entropy 2020, 22, 1047 11 of 13

By Lemma 3,

ν n νn 1 − z 2 lim z 2 · E~x exp [I, φk(t), xj(t)] [D, F,~x +~y, w~ ] n→∞ ∑ ∑ 2 j=1 k=1 Z ν n νn 1 − z 2 = lim z 2 E~x exp [I, φk(t), xj(t)] n→∞ ν[ ] 2 ∑ ∑ L2 0,T j=1 k=1 ν · exp i ∑[I, vj(t), xj(t)] j=1 ν ν · i ∑[ I, vj(t), wj(t)] · exp i ∑[ I, vj(t), yj(t)] d f (~v) j=1 j=1 ν m ν νn Z νn z − 1 Z T 1 Z T 2 − 2 2 2 = lim z z · exp [ φk(s) vj(s) ds ] − vj (t)dt n→∞ ν[ ] 2 z ∑ ∑ 2 ∑ L2 0,T j=1 k=1 0 j=1 0 ν ν · i ∑[ I, vj(t), wj(t)] · exp i ∑[ I, vj(t), yj(t)] d f (~v) j=1 j=1 Z ν ν n Z T 1 2 = lim i [I, vj(t), wj(t)] · exp − [ φk(s) vj(s) ds ] n→∞ ν[ ] ∑ 2 z ∑ ∑ L2 0,T j=1 j=1 k=1 0 ν n Z T ν Z T 1 2 1 2 · exp ∑ ∑ [ φk(s) vj(s) ds ] − ∑ vj (t) dt 2 j=1 k=1 0 2 j=1 0 ν · exp i ∑[ I, vj(t), yj(t)] d f (~v) j=1 Z ν ν ν 1 2 = i [ I, vj(t), wj(t)] · exp − ||vj||2 + i [ I, vj(t), yj(t)] d f (~v) ν[ ] ∑ z ∑ ∑ L2 0,T j=1 2 j=1 j=1 Z ν ν − 1 = i [ I, vj(t), wj(t)] · E~x exp iz 2 [ I, vj(t), xj(t)] ν[ ] ∑ ∑ L2 0,T j=1 j=1 ν · exp i ∑[ I, vj(t), yj(t)] d f (~v) j=1 Z ν = E~x i [ I, vj(t), wj(t)] ν[ ] ∑ L2 0,T j=1 ν ν − 1 · exp iz 2 ∑[ I, vj(t), xj(t) ] + i ∑[ I, vj(t), yj(t)] d f (~v) j=1 j=1 − 1 = E~x [D, F, z 2 ~x +~y, w~ ] anwz ~ ~ ~ = E~x [D, F, x + y, w ] . (43)

Theorem 7. For real ρ > 0, E~x [D, F, ρ~x +~y, w~ ]

2 ν m −νn ρ − 1 2 = lim ρ · E~x exp [I, φk(t), xj(t)] [D, F,~x +~y, w~ ] . (44) n→∞ 2 ∑ ∑ 2ρ j=1 k=1 Entropy 2020, 22, 1047 12 of 13

Proof. For real λ > 0, anwλ ~ ~ ~ E~x [D, F, x + y, w ] − 1 = E~x [D, F, λ 2 ~x +~y, w~ ]

ν m νn 1 − λ 2 = lim λ 2 · E~x exp [I, φk(t), xj(t)] [D, F,~x +~y, w~ ] . (45) n→∞ ∑ ∑ 2 j=1 k=1

Taking λ = ρ−2, Equation (44) holds.

Theorem 8. an fq ~ ~ ~ E~x [D, F, x + y, w ] ν m νn 1 − λn 2 = lim λn 2 · E~x exp [I, φk(t), xj(t)] [D, F,~x +~y, w~ ] (46) n→∞ ∑ ∑ 2 j=1 k=1 whenever {λn} → − iq through C+.

Proof. an fq ~ ~ ~ E~x [D, F, x + y, w ] anw = lim E λn [D, F,~x +~y, w~ ] n→∞ ~x ν m νn 1 − λn 2 = lim λn 2 · E~x exp [I, φk(t), xj(t)] [D, F,~x +~y, w~ ] (47) n→∞ ∑ ∑ 2 j=1 k=1 whenever {λn} → − 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).

Funding: Fund of this paper was supported by NRF-2017R1A6A3A11030667. Conﬂicts of Interest: The author declares no conﬂict of interest.

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