• Title, Summary, Keyword: function space integral

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Sequential operator-valued function space integral as an $L({L_p},{L_p'})$ theory

  • Ryu, K.S.
    • Journal of the Korean Mathematical Society
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    • v.31 no.3
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    • pp.375-391
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    • 1994
  • In 1968k Cameron and Storvick introduced the analytic and the sequential operator-valued function space integral [2]. Since then, the theo교 of the analytic operator-valued function space integral has been investigated by many mathematicians - Cameron, Storvick, Johnson, Skoug, Lapidus, Chang and author etc. But there are not that many papers related to the theory of the sequential operator-valued function space integral. In this paper, we establish the existence of the sequential operator-valued function space integral as an operator from $L_p$ to $L_p'(1 and investigated the integral equation related to this integral.

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THE PARTIAL DIFFERENTIAL EQUATION ON FUNCTION SPACE WITH RESPECT TO AN INTEGRAL EQUATION

  • Chang, Seung-Jun;Lee, Sang-Deok
    • The Pure and Applied Mathematics
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    • v.4 no.1
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    • pp.47-60
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    • 1997
  • In the theory of the conditional Wiener integral, the integrand is a functional of the standard Wiener process. In this paper we consider a conditional function space integral for functionals of more general stochastic process and the generalized Kac-Feynman integral equation. We first show that the existence of a partial differential equation. We then show that the generalized Kac-Feynman integral equation is equivalent to the partial differential equation.

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STABILITY THEOREMS OF THE OPERATOR-VALUED FUNCTION SPACE INTEGRAL ON $C_0(B)$

  • Ryu, K.-S;Yoo, S.-C
    • Bulletin of the Korean Mathematical Society
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    • v.37 no.4
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    • pp.791-802
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    • 2000
  • In 1968, Cameron and Storvick introduce the definition and the theories of the operator-valued function space integral. Since then, the stability theorems of the integral was developed by Johnson, Skoug, Chang etc [1, 2, 4, 5]. Recently, the authors establish the existence theorem of the operator-valued function space [8]. In this paper, we will prove the stability theorems of the operator-valued function space integral over paths in abstract Wiener space $C_0(B)$.

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AN OPERATOR VALUED FUNCTION SPACE INTEGRAL OF FUNCTIONALS INVOLVING DOUBLE INTEGRALS

  • Kim, Jin-Bong;Ryu, Kun-Sik
    • Communications of the Korean Mathematical Society
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    • v.12 no.2
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    • pp.293-303
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    • 1997
  • The existence theorem for the operator valued function space integral has been studied, when the wave function was in $L_1(R)$ class and the potential energy function was represented as a double integra [4]. Johnson and Lapidus established the existence theorem for the operator valued function space integral, when the wave function was in $L_2(R)$ class and the potential energy function was represented as an integral involving a Borel measure [9]. In this paper, we establish the existence theorem for the operator valued function we establish the existence theorem for the operator valued function space integral as an operator from $L_1(R)$ to $L_\infty(R)$ for certain potential energy functions which involve double integrals with some Borel measures.

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A GENERALIZED SEQUENTIAL OPERATOR-VALUED FUNCTION SPACE INTEGRAL

  • Chang, Kun-Soo;Kim, Byoung-Soo;Park, Cheong-Hee
    • Journal of the Korean Mathematical Society
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    • v.40 no.1
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    • pp.73-86
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    • 2003
  • In this paper, we define a generalized sequential operator-valued function space integral by using a generalized Wiener measure. It is an extention of the sequential operator-valued function space integral introduced by Cameron and Storvick. We prove the existence of this integral for functionals which involve some product Borel measures.

ANALYTIC OPERATOR-VALUED FUNCTION SPACE INTEGRAL REPRESENTED AS THE BOCHNER INTEGRAL:AN$L(L_2)$ THEORY

  • Chang, Kun-Soo;Park, Ki-Seong;Ryu, Kun-Sik
    • Communications of the Korean Mathematical Society
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    • v.9 no.3
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    • pp.599-606
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    • 1994
  • In [1], Cameron and Storvick introduced the analytic operator-valued function space integral. Johnson and Lapidus proved that this integral can be expressed in terms of an integral of operator-valued functions [6]. In this paper, we find some operator-valued Bochner integrable functions and prove that the analytic operator-valued function space integral of a certain function is represented as the Bochner integral of operator-valued functions on some conditions.

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PARTS FORMULAS INVOLVING CONDITIONAL INTEGRAL TRANSFORMS ON FUNCTION SPACE

  • Kim, Bong Jin;Kim, Byoung Soo
    • Korean Journal of Mathematics
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    • v.22 no.1
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    • pp.57-69
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    • 2014
  • We obtain a formula for the conditional Wiener integral of the first variation of functionals and establish several integration by parts formulas of conditional Wiener integrals of functionals on a function space. We then apply these results to obtain various integration by parts formulas involving conditional integral transforms and conditional convolution products on the function space.

GENERALIZED ANALYTIC FEYNMAN INTEGRAL VIA FUNCTION SPACE INTEGRAL OF BOUNDED CYLINDER FUNCTIONALS

  • Chang, Seung-Jun;Choi, Jae-Gil;Chung, Hyun-Soo
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.3
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    • pp.475-489
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    • 2011
  • In this paper, we use a generalized Brownian motion to define a generalized analytic Feynman integral. We then obtain some results for the generalized analytic Feynman integral of bounded cylinder functionals of the form F(x) = $\hat{v}$(($g_1,x)^{\sim}$,..., $(g_n,x)^{\sim}$) defined on a very general function space $C_{a,b}$[0,T]. We also present a change of scale formula for function space integrals of such cylinder functionals.

A bounded convergence theorem for the operator-valued feynman integral

  • Ahn, Byung-Moo
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.465-475
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    • 1996
  • Fix t > 0. Denote by $C^t$ the space of $R$-valued continuous functions x on [0,t]. Let $C_0^t$ be the Wiener space - $C_0^t = {x \in C^t : x(0) = 0}$ - equipped with Wiener measure m. Let F be a function from $C^t to C$.

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