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http://dx.doi.org/10.4134/JKMS.2012.49.5.1065

GENERALIZED FOURIER-FEYNMAN TRANSFORM AND SEQUENTIAL TRANSFORMS ON FUNCTION SPACE  

Choi, Jae-Gil (Department of Mathematics Dankook University)
Chang, Seung-Jun (Department of Mathematics Dankook University)
Publication Information
Journal of the Korean Mathematical Society / v.49, no.5, 2012 , pp. 1065-1082 More about this Journal
Abstract
In this paper we first investigate the existence of the generalized Fourier-Feynman transform of the functional F given by $$F(x)={\hat{\nu}}((e_1,x)^{\sim},{\ldots},(e_n,x)^{\sim})$$, where $(e,x)^{\sim}$ denotes the Paley-Wiener-Zygmund stochastic integral with $x$ in a very general function space $C_{a,b}[0,T]$ and $\hat{\nu}$ is the Fourier transform of complex measure ${\nu}$ on $B({\mathbb{R}}^n)$ 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;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
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1 T. Huffman, C. Park, and D. Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc. 347 (1995), no. 2, 661-673.   DOI
2 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.   DOI
3 T. Huffman, C. Park, and D. Skoug, Convolution and Fourier-Feynman transforms, Rocky Mountain J. Math. 27 (1997), no. 3, 827-841.
4 G. W. Johnson and D. L. Skoug, Scale-invariant measurability in Wiener space, Pacific J. Math. 83 (1979), no. 1, 157-176.   DOI
5 G. W. Johnson and D. L. Skoug, An $L_{p}$ analytic Fourier-Feynman transform, Michigan Math. J. 26 (1979), no. 1, 103-127.   DOI
6 H. L. Royden, Real Analysis (Third edition), Macmillan, 1988.
7 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.   DOI   ScienceOn
8 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.
9 J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973.
10 M. D. Brue, A functional transform for Feynman integrals similar to the Fourier transform, Thesis, University of Minnesota, Minneapolis, 1972.
11 R. H. Cameron and D. A. Storvick, An $L_{2}$ analytic Fourier-Feynman transform, Michigan Math. J. 23 (1976), no. 1, 1-30.   DOI
12 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.   과학기술학회마을   DOI   ScienceOn
13 S. J. Chang and D. M. Chung, Conditional function space integrals with applications, Rocky Mountain J. Math. 26 (1996), no. 1, 37-62.   DOI
14 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.   과학기술학회마을
15 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.   DOI   ScienceOn
16 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.   DOI   ScienceOn
17 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.   DOI
18 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.   DOI   ScienceOn