A TIME-INDEPENDENT CONDITIONAL FOURIER-FEYNMAN TRANSFORM AND CONVOLUTION PRODUCT ON AN ANALOGUE OF WIENER SPACE

• Journal title : Honam Mathematical Journal
• Volume 35, Issue 2,  2013, pp.179-200
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2013.35.2.179
Title & Authors
A TIME-INDEPENDENT CONDITIONAL FOURIER-FEYNMAN TRANSFORM AND CONVOLUTION PRODUCT ON AN ANALOGUE OF WIENER SPACE
Cho, Dong Hyun;

Abstract
Let $\small{C[0,t]}$ denote the function space of all real-valued continuous paths on $\small{[0,t]}$. Define $\small{X_n:C[0,t]{\rightarrow}\mathbb{R}^{n+1}}$ by $\small{Xn(x)=(x(t_0),x(t_1),{\cdots},x(t_n))}$, where $\small{0=t_0}$ < $\small{t_1}$ < $\small{{\cdots}}$ < $\small{t_n}$ < $\small{t}$ is a partition of $\small{[0,t]}$. In the present paper, using a simple formula for the conditional expectation given the conditioning function $\small{X_n}$, we evaluate the $\small{L_p(1{\leq}p{\leq}{\infty})}$-analytic conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions which have the form $\small{f((v_1,x),{\cdots},(v_r,x))\;for\;x{\in}C[0,t]}$, where {$\small{v_1,{\cdots},v_r}$} is an orthonormal subset of $\small{L_2[0,t]}$ and $\small{f{\in}L_p(\mathbb{R}^r)}$. We then investigate several relationships between the conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions.
Keywords
analogue of Wiener measure;analytic conditional Feynman integral;analytic conditional Fourier-Feynman transform;analytic conditional Wiener integral;conditional convolution product;
Language
English
Cited by
1.
RELATIONSHIPS BETWEEN INTEGRAL TRANSFORMS AND CONVOLUTIONS ON AN ANALOGUE OF WIENER SPACE,;

호남수학학술지, 2013. vol.35. 1, pp.51-71
2.
CONDITIONAL FOURIER-FEYNMAN TRANSFORMS AND CONVOLUTIONS OF UNBOUNDED FUNCTIONS ON A GENERALIZED WIENER SPACE,;

대한수학회지, 2013. vol.50. 5, pp.1105-1127
3.
CONDITIONAL INTEGRAL TRANSFORMS AND CONVOLUTIONS OF BOUNDED FUNCTIONS ON AN ANALOGUE OF WIENER SPACE,;

충청수학회지, 2013. vol.26. 2, pp.323-342
1.
CONDITIONAL FOURIER-FEYNMAN TRANSFORMS AND CONVOLUTIONS OF UNBOUNDED FUNCTIONS ON A GENERALIZED WIENER SPACE, Journal of the Korean Mathematical Society, 2013, 50, 5, 1105
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