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Let $C[0,t]$ denote the function space of all real-valued continuous paths on $[0,t]$. Define $Xn:C[0,t]{\rightarrow}\mathbb{R}^{n+1}$ and $X_{n+1}:C[0,t]{\rightarrow}\mathbb{R}^{n+2}$ by $X_n(x)=(x(t_0),x(t_1),{\cdots},x(t_n))$ and $X_{n+1}(x)=(x(t_0),x(t_1),{\cdots},x(t_n),x(t_{n+1}))$, where $0=t_0$ < $t_1$ < ${\cdots}$ < $t_n$ < $t_{n+1}=t$. In the present paper, using simple formulas for the conditional expectations with the conditioning functions $X_n$ and $X_{n+1}$, we evaluate the $L_p(1{\leq}p{\leq}{\infty})$-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the functions which have the form $${\int}_{L_2[0,t]}{{\exp}\{i(v,x)\}d{\sigma}(v)}{{\int}_{\mathbb{R}^r}}\;{\exp}\{i{\sum_{j=1}^{r}z_j(v_j,x)\}dp(z_1,{\cdots},z_r)$$ for $x{\in}C[0,t]$, where $\{v_1,{\cdots},v_r\}$ is an orthonormal subset of $L_2[0,t]$ and ${\sigma}$ and ${\rho}$ are the complex Borel measures of bounded variations on $L_2[0,t]$ and $\mathbb{R}^r$, respectively. We then investigate the inverse transforms of the function with their relationships and finally prove that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions, can be expressed in terms of the products of the conditional Fourier-Feynman transforms of each function.
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참고문헌 (14)
참고문헌 신청
R. H. Cameron / 1980 / Some Banach algebras of analytic Feynman integrable functionals, In Lecture Notes in Mathematics / Springer
K. S. Chang / 2003 / Conditional Fourier-Feynman transform and convolution product over Wiener paths in abstract Wiener space / Integral Transforms Spec. Funct 14(3):217~235
S. J. Chang / 2000 / The e®ect of drift on conditional Fourier-Feynman transforms and conditional convolution products / Int. J. Appl. Math 2(4):505~527
조동현 / 2013 / A TIME-INDEPENDENT CONDITIONAL FOURIER-FEYNMAN TRANSFORM AND CONVOLUTION PRODUCT ON AN ANALOGUE OF WIENER SPACE / 호남수학학술지 35(2):179~200
D. H. Cho / 2012 / A time-dependent conditional Fourier-Feynman transform and convolution product on an analogue of Wiener space / Houston J. Math
K. S. Chang / 2003 / Conditional Fourier-Feynman transform and convolution product over Wiener paths in abstract Wiener space / Integral Transforms Spec. Funct 14(3):217~235
S. J. Chang / 2000 / The e®ect of drift on conditional Fourier-Feynman transforms and conditional convolution products / Int. J. Appl. Math 2(4):505~527
조동현 / 2013 / A TIME-INDEPENDENT CONDITIONAL FOURIER-FEYNMAN TRANSFORM AND CONVOLUTION PRODUCT ON AN ANALOGUE OF WIENER SPACE / 호남수학학술지 35(2):179~200
D. H. Cho / 2012 / A time-dependent conditional Fourier-Feynman transform and convolution product on an analogue of Wiener space / Houston J. Math
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