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GENERALIZED FOURIER-FEYNMAN TRANSFORM AND SEQUENTIAL TRANSFORMS ON FUNCTION SPACE

  • Received : 2011.07.29
  • Published : 2012.09.01

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

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. https://doi.org/10.1307/mmj/1029001617
  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. https://doi.org/10.4134/CKMS.2011.26.2.273
  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. https://doi.org/10.1090/S0002-9947-03-03256-2
  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. https://doi.org/10.1216/RMJ-2010-40-3-761
  7. 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
  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. https://doi.org/10.1007/s00041-009-9076-y
  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. https://doi.org/10.1080/1065246031000074425
  10. T. Huffman, C. Park, and D. Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc. 347 (1995), no. 2, 661-673. https://doi.org/10.1090/S0002-9947-1995-1242088-7
  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. https://doi.org/10.1307/mmj/1029005461
  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. https://doi.org/10.2140/pjm.1979.83.157
  14. G. W. Johnson and D. L. Skoug, An $L_{p}$ analytic Fourier-Feynman transform, Michigan Math. J. 26 (1979), no. 1, 103-127. https://doi.org/10.1307/mmj/1029002166
  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. https://doi.org/10.1216/rmjm/1181069848
  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.

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