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CONDITIONAL GENERALIZED FOURIER-FEYNMAN TRANSFORM OF FUNCTIONALS IN A FRESNEL TYPE CLASS

Chang, Seung-Jun

  • Received : 2010.04.03
  • Published : 2011.04.30

Abstract

In this paper we dene the concept of a conditional generalized Fourier-Feynman transform on very general function space $C_{a,b}$[0, T]. We then establish the existence of the conditional generalized Fourier-Feynman transform for functionals in a Fresnel type class. We also obtain several results involving the conditional transform. Finally we present functionals to apply our results. The functionals arise naturally in Feynman integration theories and quantum mechanics.

Keywords

generalized Brownian motion;Fresnel type class;generalized Fourier-Feynman transform;conditional generalized Feynman integral;conditional generalized Fourier-Feynman transform

References

  1. R. H. Cameron and D. A. Storvick, Some Banach algebras of analytic Feynman integrable functionals, Analytic Functions, Kozubnik 1979 (Proc. Seventh Conf., Kozubnik, 1979), pp. 18-67, Lecture Notes in Math., 798, Springer, Berlin-New York, 1980.
  2. S. J. Chang and D. M. Chung, Conditional function space integrals with applications, Rocky Mountain J. Math. 26 (1996), no. 1, 37-62. https://doi.org/10.1216/rmjm/1181072102
  3. S. J. Chang and D. Skoug, Parts formulas involving conditional Feynman integrals, Bull. Austral. Math. Soc. 65 (2002), no. 3, 353-369. https://doi.org/10.1017/S0004972700020402
  4. S. J. Chang and D. Skoug, Generalized Fourier-Feynman transforms and a first variation on function space, Integral Transforms Spec. Funct. 14 (2003), no. 5, 375-393. https://doi.org/10.1080/1065246031000074425
  5. S. J. Chang and J. G. Choi, Conditional generalized Fourier-Feynman transform and conditional convolution product on a Banach algebra, Bull. Korean Math. Soc. 41 (2004), no. 1, 73-93. https://doi.org/10.4134/BKMS.2004.41.1.073
  6. S. J. Chang and J. G. Choi, Multiple $L_p$ analytic generalized Fourier-Feynman transform on the Banach algebra, Commun. Korean Math. Soc. 19 (2004), no. 1, 93-111. https://doi.org/10.4134/CKMS.2004.19.1.093
  7. 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. https://doi.org/10.1090/S0002-9947-03-03256-2
  8. S. J. Chang, J. G. Choi, and D. Skoug, Parts formulas involving conditional generalized Feynman integrals and conditional generalized Fourier-Feynman transforms on function space, Integral Transforms Spec. Funct. 15 (2004), no. 6, 491-512. https://doi.org/10.1080/1065246042000271983
  9. S. J. Chang, J. G. Choi, and D. Skoug, Evaluation formulas for conditional function space integrals. I, Stoch. Anal. Appl. 25 (2007), no. 1, 141-168. https://doi.org/10.1080/07362990601052185
  10. S. J. Chang, J. G. Choi, and D. Skoug, Simple formulas for conditional function space integrals and applications, Integration: Mathematical Theory and Applications 1 (2008), 1-20.
  11. 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. https://doi.org/10.1007/s00041-009-9076-y
  12. S. J. Chang, J. G. Choi, and S. D. Lee, A Fresnel type class on function space, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 16 (2009), no. 1, 107-119.
  13. D. M. Chung, Conditional analytic Feynman integrals on abstract Wiener spaces, Proc. Amer. Math. Soc. 112 (1991), no. 2, 479-488. https://doi.org/10.1090/S0002-9939-1991-1060719-3
  14. D. M. Chung and D. L. Skoug, Conditional analytic Feynman integrals and a related Schrodinger integral equation, SIAM J. Math. Anal. 20 (1989), no. 4, 950-965. https://doi.org/10.1137/0520064
  15. G. W. Johnson and D. L. Skoug, Notes on the Feynman integral. III. The Schroedinger equation, Pacific J. Math. 105 (1983), no. 2, 321-358. https://doi.org/10.2140/pjm.1983.105.321
  16. G. W. Johnson and M. L. Lapidus, The Feynman Integral and Feynman's Operational Calculus, Clarendon Press, Oxford, 2000.
  17. C. Park and D. Skoug, A simple formula for conditional Wiener integrals with applications , Pacific J. Math. 135 (1988), no. 2, 381-394. https://doi.org/10.2140/pjm.1988.135.381
  18. C. Park and D. Skoug, Conditional Fourier-Feynman transforms and conditional convolution products, J. Korean Math. Soc. 38 (2001), no. 1, 61-76.
  19. 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.
  20. J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973.
  21. J. Yeh, Inversion of conditional Wiener integrals, Pacific J. Math. 59 (1975), no. 2, 623-638. https://doi.org/10.2140/pjm.1975.59.623

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