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

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Partial Derivative Approach to the Integral Transform for the Function Space in the Banach Algebra entropy 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 rE 2 M for all r > 0 and a scale invariant measurable set N is said to be scale-invariant null provided m(rN) = 0 for each r > 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+ = fljRe(l) > 0g and C+ = fljRe(l) ≥ 0g. For real l > 0, − 1 JF(l) = Ex F(l 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 2 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 l 2 C+ and for s − a.e.y 2 C0[0, T], let anwl ( Tl(F) )(y) = Ex [F( x + y )]. (4) Definition 2. For the L1-analytic Fourier–Feynmann transform, see Definition 2 in [5]: (1) anwl an fq ( Tq (F) )(y) = lim Ex [F( x + y )] = Ex [F( x + y )] , (5) l!−iq whenever l ! −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 2 C0[0, T] which is defined by the partial derivative as ¶ dF(xjw) = F(x + hw)j . (6) ¶h h=0 We will denote it by [D, F, x, w]. Remark 1. For a 2 C+ and b 2 R, Z r p b2 expf−au2 + ibug du = expf− g . (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 jjvjj dj f j(v) < ¥. L2[0,T] 2 Suppose that formulas in this section hold for s − a.e.w 2 C0[0, T] and for s − a.e.y 2 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 )j ¶h h=0 ¶ Z = exp i h [I, v(t), w(t)] + i [I, v(t), x(t)] d f (v) jh=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)] dj f j(v) . (11) L2[0,T] Then Z Ew [I, v(t), w(t)] dj f j(v) L2[0,T] Z = Ew [I, v(t), w(t)] dj f j(v) L2[0,T] Z 1 Z +¥ u2 = j j − j j( ) q u exp 2 du d f v L2[0,T] 2 −¥ 2jjvjj 2p jvjj2 2 r 2 Z = jjvjj2 dj f j(v) < ¥ (12) p L2[0,T] R where Ew F(w) = F(w) dm(w). So, C0[0,T] Z [I, v(t), w(t)] dj f j(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] = dF(x + yjw). Entropy 2020, 22, 1047 4 of 13 Proof. [1]. For z 2 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 − jjvjj2 + 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 ( − jjvjj2 + 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 l 2 C+, n 1 − l 2 exp ∑ [I, fk(t), x(t)] · [D, F, x + y, w] (17) 2 k=1 is a Wiener integrable function of x 2 C0[0, T]. Proof. n 1 − l 2 Ex exp ∑ [I, fk(t), x(t)] [D, F, x + y, w] 2 k=1 n 1 − l 2 = Ex exp ∑ [I, fk(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 − l 2 = i [I, v(t), w(t)] · Ex exp ∑ [I, fk(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 − l 2 Ex exp ∑ [I, fk(t), x(t)] · exp i [I, v(t), x(t)] + i [I, v(t), y(t)] 2 k=1 n 1 − l 2 ≤ Ex exp ∑ [I, fk(t), x(t)] 2 k=1 Z n − n l 2 = (2p) 2 exp − u d~u n ∑ k R 2 k=1 − n 2p n = (2p) 2 · ( ) 2 l − n = l 2 . (19) Therefore, Z n 1 − l 2 i [I, v(t), w(t)] · Ex exp ∑ [I, fk(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 ≤ l 2 · [I, v(t), w(t)] dj f j(v) . (20) L2[0,T] By the Wiener integration theorem, Z Ew [I, v(t), w(t)] dj f j(v) L2[0,T] Z 1 Z +¥ u2 = · j j · − j j( ) q u exp 2 du d f v L2[0,T] 2 −¥ 2jjvjj 2 pjjvjj2 2 2 1 Z = ( ) 2 jjvjj2 dj f j(v) < ¥ . (21) p L2[0,T] m Lemma 3 (Ref. [12]). Let ffjgj=1 be an orthonormal set in L2[0, T].
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