GENERALIZED FOURIER-FEYNMAN TRANSFORM AND SEQUENTIAL TRANSFORMS ON FUNCTION SPACE

Title & Authors
GENERALIZED FOURIER-FEYNMAN TRANSFORM AND SEQUENTIAL TRANSFORMS ON FUNCTION SPACE
Choi, Jae-Gil; Chang, Seung-Jun;

Abstract
In this paper we first investigate the existence of the generalized Fourier-Feynman transform of the functional F given by $\small{F(x)={\hat{\nu}}((e_1,x)^{\sim},{\ldots},(e_n,x)^{\sim})}$, where $\small{(e,x)^{\sim}}$ denotes the Paley-Wiener-Zygmund stochastic integral with $\small{x}$ in a very general function space $\small{C_{a,b}[0,T]}$ and $\small{\hat{\nu}}$ is the Fourier transform of complex measure $\small{{\nu}}$ on $\small{B({\mathbb{R}}^n)}$ with finite total variation. We then define two sequential transforms. Finally, we establish that the one is to identify the generalized Fourier-Feynman transform and the another transform acts like an inverse generalized Fourier-Feynman transform.
Keywords
generalized Brownian motion process;Paley-Wiener-Zygmund stochastic integral;cylinder functional;generalized Fourier-Feynman transform;sequential P-transform;sequential N-transform;
Language
English
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3.
L2-sequential transforms on function space, Journal of Mathematical Analysis and Applications, 2015, 421, 1, 625
4.
Generalized Analytic Fourier-Feynman Transform of Functionals in a Banach AlgebraℱA1,A2a,b, Journal of Function Spaces and Applications, 2013, 2013, 1
References
1.
M. D. Brue, A functional transform for Feynman integrals similar to the Fourier transform, Thesis, University of Minnesota, Minneapolis, 1972.

2.
R. H. Cameron and D. A. Storvick, An $L_{2}$ analytic Fourier-Feynman transform, Michigan Math. J. 23 (1976), no. 1, 1-30.

3.
S. J. Chang, Conditional generalized Fourier-Feynman transform of functionals in a Fresnel type class, Commun. Korean Math. Soc. 26 (2011), no. 2, 273-289.

4.
S. J. Chang, J. G. Choi, and H. S. Chung, Generalized analytic Feynman integral via function space integral of bounded cylinder functionals, Bull. Korean Math. Soc. 48 (2011), no. 3, 475{489.

5.
S. J. Chang, J. G. Choi, and D. Skoug, Integration by parts formulas involving gen- eralized Fourier-Feynman transforms on function space, Trans. Amer. Math. Soc. 355 (2003), no. 7, 2925-2948.

6.
S. J. Chang, J. G. Choi, and D. Skoug, Generalized Fourier-Feynman transforms, convolution products, and rst variations on function space, Rocky Mountain J. Math. 40 (2010), no. 3, 761-788.

7.
S. J. Chang and D. M. Chung, Conditional function space integrals with applications, Rocky Mountain J. Math. 26 (1996), no. 1, 37-62.

8.
S. J. Chang, H. S. Chung, and D. Skoug, Integral transforms of functionals in $L^{2}$($C_{a,b}$[0, T]), J. Fourier Anal. Appl. 15 (2009), no. 4, 441-462.

9.
S. J. Chang and D. Skoug, Generalized Fourier-Feynman transforms and a rst variation on function space, Integral Transforms Spec. Funct. 14 (2003), no. 5, 375-393.

10.
T. Huffman, C. Park, and D. Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc. 347 (1995), no. 2, 661-673.

11.
T. Huffman, C. Park, and D. Skoug, Convolutions and Fourier-Feynman transforms of functionals involving multiple integrals, Michigan Math. J. 43 (1996), no. 2, 247-261.

12.
T. Huffman, C. Park, and D. Skoug, Convolution and Fourier-Feynman transforms, Rocky Mountain J. Math. 27 (1997), no. 3, 827-841.

13.
G. W. Johnson and D. L. Skoug, Scale-invariant measurability in Wiener space, Pacific J. Math. 83 (1979), no. 1, 157-176.

14.
G. W. Johnson and D. L. Skoug, An $L_{p}$ analytic Fourier-Feynman transform, Michigan Math. J. 26 (1979), no. 1, 103-127.

15.
H. L. Royden, Real Analysis (Third edition), Macmillan, 1988.

16.
D. Skoug and D. Storvick, A survey of results involving transforms and convolutions in function space, Rocky Mountain J. Math. 34 (2004), no. 3, 1147-1175.

17.
J. Yeh, Singularity of Gaussian measures on function space induced by Brownian motion processes with non-stationary increments, Illinois J. Math. 15 (1971), 37-46.

18.
J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973.