GENERALIZED FOURIER-WIENER FUNCTION SPACE TRANSFORMS

Title & Authors
GENERALIZED FOURIER-WIENER FUNCTION SPACE TRANSFORMS
Chang, Seung-Jun; Chung, Hyun-Soo;

Abstract
In this paper, we define generalized Fourier-Hermite functionals on a function space $\small{C_{a,b}[0,\;T]}$ to obtain a complete orthonormal set in $\small{L_2(C_{a,b}[0,\;T])}$ where $\small{C_{a,b}[0,\;T]}$ is a very general function space. We then proceed to give a necessary and sufficient condition that a functional F in $\small{L_2(C_{a,b}[0,\;T])}$ has a generalized Fourier-Wiener function space transform $\small{{\cal{F}}_{\sqrt{2},i}(F)}$ also belonging to $\small{L_2(C_{a,b}[0,\;T])}$.
Keywords
generalized Brownian motion process;generalized Hermite function;generalized Fourier-Hermite coefficient;generalized Fourier-Wiener function space transform;
Language
English
Cited by
1.
INTEGRAL TRANSFORMS AND INVERSE INTEGRAL TRANSFORMS WITH RELATED TOPICS ON FUNCTION SPACE I,;;

한국수학교육학회지시리즈B:순수및응용수학, 2009. vol.16. 4, pp.369-382
2.
SERIES EXPANSIONS OF THE ANALYTIC FEYNMAN INTEGRAL FOR THE FOURIER-TYPE FUNCTIONAL,;;;

한국수학교육학회지시리즈B:순수및응용수학, 2012. vol.19. 2, pp.87-102
3.
REFLECTION PRINCIPLES FOR GENERAL WIENER FUNCTION SPACES,;;

대한수학회지, 2013. vol.50. 3, pp.607-625
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SERIES EXPANSIONS OF THE ANALYTIC FEYNMAN INTEGRAL FOR THE FOURIER-TYPE FUNCTIONAL, The Pure and Applied Mathematics, 2012, 19, 2, 87
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References
1.
R. H. Cameron, Some examples of Fourier-Wiener transforms of analytic functionals, Duke Math. J. 12 (1945), 485–488.

2.
R. H. Cameron and W. T. Martin, Fourier-Wiener transforms of analytic functionals, Duke Math. J. 12 (1945), 489–507.

3.
R. H. Cameron and W. T. Martin, Fourier-Wiener transforms of functionals belonging to $L_2$ over the space Ć, Duke Math. J. 14 (1947), 99–107.

4.
R. H. Cameron and W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Ann. of Math. 48 (1947), 385–392.

5.
R. H. Cameron and D. A. Storvick, An $L_2$ analytic Fourier-Feynman transform, Michigan Math. J. 23 (1976), no. 1, 1–30.

6.
K. S. Chang, B. S. Kim, and I. Yoo, Integral transform and convolution of analytic functionals on abstract Wiener space, Numer. Funct. Anal. Optim. 21 (2000), no. 1-2, 97–105.

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

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

10.
B. S. Kim and D. Skoug, Integral transforms of functionals in $L_{2}$($C_{0}$[0, T]), Rocky Mountain J. Math. 33 (2003), no. 4, 1379–1393.

11.
Y. J. Lee, Integral transforms of analytic functions on abstract Wiener spaces, J. Funct. Anal. 47 (1982), no. 2, 153–164.

12.
Y. J. Lee, Unitary operators on the space of $L^2$-functions over abstract Wiener spaces, Soochow J. Math. 13 (1987), no. 2, 165–174.

13.
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.

14.
E. Nelson, Dynamical Theories of Brownian Motion (2nd edition), Math. Notes, Princeton University Press, Princeton, 1967.

15.
J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973.

16.
J. Yeh, Singularity of Gaussian measures on function spaces induced by Brownian motion processes with non-stationary increments, Illinois J. Math. 15 (1971), 37–46.