• Korean Journal of Mathematics
• /
• v.11 no.1
• /
• pp.17-30
• /
• 2003

In this paper, we use a generalized Brownian motion process to define a translation theorem. First we establish the translation theorem for function space integrals. We then obtain the general translation theorem for functionals on function space.

• Journal of the Chungcheong Mathematical Society
• /
• v.26 no.4
• /
• pp.735-742
• /
• 2013

In this note, we prove the translation theorem for the generalized analogue of Wiener measure space and we show some properties of the generalized analogue of Wiener measure from it.

• Journal of the Korean Mathematical Society
• /
• v.39 no.5
• /
• pp.801-819
• /
• 2002

In this note, we establish a translation theorem in an analogue of Wiener space (C[0,t],$\omega$$\phi$) and find formulas for the conditional $\omega$$\phi$-integral given by the condition X(x) ＝ (x(to), x(t$_1$),…, x(t$_{n}$)) which is the generalization of Chang and Chang's results in 1984. Moreover, we prove a translation theorem for the conditional $\omega$$\phi$-integral.l.

• Journal of the Korean Mathematical Society
• /
• v.57 no.3
• /
• pp.655-668
• /
• 2020

In this paper, we establish the generalized Cameron-Storvick type theorem on function space. We then give relationships involving the generalized Cameron-Storvick type theorem, modified generalized integral transform and modified convolution product. A motivation of studying the generalized Cameron-Storvick type theorem is to generalize formulas and results with respect to the modified generalized integral transform on function space. From the some theories and formulas in the functional analysis, we can obtain some formulas with respect to the translation theorem of exponential functionals.

• Honam Mathematical Journal
• /
• v.27 no.2
• /
• pp.225-232
• /
• 2005

We obtain the approximation order to a smooth function on a compact subset of $\mathbb{R}$ by generalized translation networks. In our study, the activation function is infinitely many times continuously differentiable function but it does not have special properties around ${\infty}$ and $-{\infty}$ like a sigmoidal activation function. Using the Jackson's Theorem, we get the approximation order. Especially, we obtain the approximation order by a neural network with a fixed threshold.

• Journal of the Korean Mathematical Society
• /
• v.53 no.5
• /
• pp.991-1017
• /
• 2016

In this paper we define a generalized analytic Fourier-Feynman transform associated with Gaussian process on the function space $C_{a,b}[0,T]$. We establish the existence of the generalized analytic Fourier-Feynman transform for certain bounded functionals on $C_{a,b}[0,T]$. We then proceed to establish a translation theorem for the generalized transform associated with Gaussian process.

• Journal of the Korean Mathematical Society
• /
• v.53 no.6
• /
• pp.1261-1273
• /
• 2016

In this paper, we analyze the necessary and sufficient condition introduced in [5]: that a functional F in $L^2(C_{a,b}[0,T])$ has an integral transform ${\mathcal{F}}_{{\gamma},{\beta}}F$, also belonging to $L^2(C_{a,b}[0,T])$. We then establish the inverse integral transforms of the functionals in $L^2(C_{a,b}[0,T])$ and then examine various properties with respect to the inverse integral transforms via the translation theorem. Several possible outcomes are presented as remarks. Our approach is a new method to solve some difficulties with respect to the inverse integral transform.

• Bulletin of the Korean Mathematical Society
• /
• v.59 no.5
• /
• pp.1093-1103
• /
• 2022

In this paper, we establish an evaluation formula to calculate the Wiener integral of polynomials in terms of natural dual pairings on abstract Wiener spaces (H, B, 𝜈). To do this we first derive a translation theorem for the Wiener integral of functionals associated with operators in 𝓛(B), the Banach space of bounded linear operators from B to itself. We then apply the translation theorem to establish an integration by parts formula for the Wiener integral of functionals combined with operators in 𝓛(B). We finally apply this parts formula to evaluate the Wiener integral of certain polynomials in terms of natural dual pairings.

• Bulletin of the Korean Mathematical Society
• /
• v.36 no.3
• /
• pp.419-439
• /
• 1999

In this paper, we first develop a general formula for evaluating conditional abstract Wiener integrals of cylinder functions. we next use our formula to evaluate the conditional abstract wiener integral of various cylinder functions and then specialize our results to conditional Yeh-Wiener integrals to show that we can obtain the corresponding results by Park and Skoug. We finally obtain a Cameron-Martin translation theorem for conditional abstract Wiener integrals.

• East Asian mathematical journal
• /
• v.14 no.2
• /
• pp.237-247
• /
• 1998

In this paper, we prove a translation theorem for the analytic Feynman integral for functions in $\mathcal{F}_{A_1,A_2}$(B) and show how this integral can be modified so as to be translation invariant.