CONDITIONAL FOURIER-FEYNMAN TRANSFORMS AND CONVOLUTIONS OF UNBOUNDED FUNCTIONS ON A GENERALIZED WIENER SPACE

Title & Authors
CONDITIONAL FOURIER-FEYNMAN TRANSFORMS AND CONVOLUTIONS OF UNBOUNDED FUNCTIONS ON A GENERALIZED WIENER SPACE
Cho, Dong Hyun;

Abstract
Let C[0, $\small{t}$] denote the function space of real-valued continuous paths on [0, $\small{t}$]. Define $\small{X_n\;:\;C[0,t}$$\small{]}$$\small{{\rightarrow}\mathbb{R}^{n+1}}$ and $\small{X_{n+1}\;:\;C[0,t}$$\small{]}$$\small{{\rightarrow}\mathbb{R}^{n+2}}$ by $X_n(x) Keywords analogue of Wiener space;analytic conditional Feynman integral;analytic conditional Fourier-Feynman transform;analytic conditional Wiener integral;conditional convolution product;Wiener space; Language English Cited by 1. A TRANSLATION THEOREM FOR THE GENERALIZED FOURIER-FEYNMAN TRANSFORM ASSOCIATED WITH GAUSSIAN PROCESS ON FUNCTION SPACE,;;; 대한수학회지, 2016. vol.53. 5, pp.991-1017 1. A TRANSLATION THEOREM FOR THE GENERALIZED FOURIER-FEYNMAN TRANSFORM ASSOCIATED WITH GAUSSIAN PROCESS ON FUNCTION SPACE, Journal of the Korean Mathematical Society, 2016, 53, 5, 991 References 1. M. D. Brue, A functional transform for Feynman integrals similar to the Fourier transform, Thesis, Univ. of Minnesota, Minneapolis, 1972. 2. R. H. Cameron and D. A. Storvick, Some Banach algebras of analytic Feynman integrable functionals, Lecture Notes in Mathematics 798, Springer, Berlin-New York, 1980. 3. K. S. Chang, D. H. Cho, B. S. Kim, T. S. Song, and I. Yoo, Conditional Fourier-Feynman transform and convolution product over Wiener paths in abstract Wiener space, Integral Transforms Spec. Funct. 14 (2003), no. 3, 217-235. 4. S. J. Chang and D. M. Chung, A class of conditional Wiener integrals, J. Korean Math. Soc. 30 (1993), no. 1, 161-172. 5. S. J. Chang and D. Skoug, The effect of drift on conditional Fourier-Feynman trans- forms and conditional convolution products, Int. J. Appl. Math. 2 (2000), no. 4, 505-527. 6. S. J. Chang and D. Skoug, The effect of drift on the Fourier-Feynman transform, the convolution product and the first variation, Panamer. Math. J. 10 (2000), no. 2, 25-38. 7. D. H. Cho, A time-independent conditional Fourier-Feynman transform and convolution product on an analogue of Wiener space, Honam Math. J. (2013), submitted. 8. D. H. Cho, A time-dependent conditional Fourier-Feynman transform and convolution product on an analogue of Wiener space, Houston J. Math. (2012), submitted. 9. D. H. Cho, Conditional integral transforms and convolutions of bounded functions on an analogue of Wiener space, J. Chungcheong Math. Soc. (2012), to appear. 10. D. H. Cho, Conditional integral transforms and conditional convolution products on a function space, Integral Transforms Spec. Funct. 23 (2012), no. 6, 405-420. 11. D. H. Cho, A simple formula for an analogue of conditional Wiener integrals and its applications II, Czechoslovak Math. J. 59 (2009), no. 2, 431-452. 12. D. H. Cho, A simple formula for an analogue of conditional Wiener integrals and its applications, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3795-3811. 13. D. H. Cho, Conditional Fourier-Feynman transform and convolution product over Wiener paths in abstract Wiener space: an$L_p$theory, J. Korean Math. Soc. 41 (2004), no. 2, 265-294. 14. D. H. Cho, B. J. Kim, and I. Yoo, Analogues of conditional Wiener integrals and their change of scale transformations on a function space, J. Math. Anal. Appl. 359 (2009), no. 2, 421-438. 15. T. Huffman, C. Park, and D. Skoug, Convolutions and Fourier-Feynman transforms of functionals involving multiple integrals, Michigan Math. J. 43 (1996), no. 2, 247-261. 16. M. K. Im and K. S. Ryu, An analogue of Wiener measure and its applications, J. Korean Math. Soc. 39 (2002), no. 5, 801-819. 17. G. W. Johnson and D. L. Skoug, The Cameron-Storvick function space integral: an L$(L_p,L_p{\prime})\$ theory, Nagoya Math. J. 60 (1976), 93-137.

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