Kyungpook Mathematical Journal

  • 제64권2호
  • /
  • Pages.219-233
  • /
  • 2024
  • /
  • 1225-6951(pISSN)
  • /
  • 0454-8124(eISSN)
경북대학교 자연과학대학 수학과 (Department of Mathematics, Kyungpook National University)
권호 표지
DOI QR코드

DOI QR Code

Generalized Fourier-Feynman Transform of Bounded Cylinder Functions on the Function Space Ca,b[0, T]

  • Jae Gil Choi (Department of Mathematics, Dankook University)
  • 투고 : 2023.08.05
  • 심사 : 2024.03.10
  • 발행 : 2024.06.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we study the generalized Fourier-Feynman transform (GFFT) for functions on the general Wiener space Ca,b[0, T]. We establish an explicit evaluation formula for the analytic GFFT of bounded cylinder functions on Ca,b[0, T]. We start by examining certain cylinder functions which belong in a Banach algebra of bounded functions on Ca,b[0, T]. We then obtain an explicit formula for the analytic GFFT of the bounded cylinder functions.

키워드

참고문헌

  1. M. D. Brue, A Functional Transform for Feynman Integrals Similar to the Fourier Transform, Ph.D. Thesis, University of Minnesota, Minneapolis(1972)
  2. R. H. Cameron and D. A. Storvick, An L2 analytic Fourier-Feynman transform, Michigan Math. J., 23(1976), 1-30.
  3. R. H. Cameron and D. A. Storvick, Analytic Feynman integral solutions of an integral equation related to the Schroedinger equation, J. Anal. Math., 38(1980), 34-66.
  4. R. H. Cameron and D. A. Storvick, Some Banach Algebras of Analytic Feynman Integrable Functionals. Analytic Functions (Proc. Seventh Conf., Kozubnik, 1979), 18-67, Lecture Notes in Math. vol. 798, Springer, Berlin(1980)
  5. K. S. Chang, G. W. Johnson and D. L. Skoug, Necessary and sufficient conditions for the Fresnel integrability of certain classes of functions, J. Korean Math. Soc., 21(1)(1984), 21-29.
  6. K. S. Chang, G. W. Johnson and D. L. Skoug, Necessary and sufficient conditions for membership in the Banach algebras S for certain classes of functions, Rend. Circ. Mat. Palermo (2) Suppl., 17(1987), 153-171.
  7. K. S. Chang, G. W. Johnson and D. L. Skoug, Functions in the Fresnel class, Proc. Amer. Math. Soc., 100(1987), 309-318.
  8. S. J. Chang and J. G. Choi, Generalized Fourier-Feynman transform and sequential transforms on function space, J. Korean Soc. Math., 49(2012), 1065-1082.
  9. 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), 2925-2948.
  10. S. J. Chang, J. G. Choi and D. Skoug, Evaluation formulas for conditional function space integrals I, Stoch. Anal. Appl., 25(2007), 141-168.
  11. S. J. Chang and D. Skoug, Generalized Fourier-Feynman transforms and a first variation on function space, Integral Transforms Spec. Funct., 14(2003), 375-393.
  12. J. G. Choi, D. Skoug and S. J. Chang, Generalized analytic Fourier-Feynman transform of functionals in a Banach algebra $F_{A_1,A_2}^{a,b}$, J. Funct. Spaces Appl., 2013(2013), Article ID 954098, 1-12.
  13. D. M. Chung and H. T. Hwang, Cylinder functions in the Fresnel class of functions on abstract Wiener spaces, Proc. Amer. Math. Soc., 115(1992), 381-388.
  14. T. Huffman, C. Park and D. Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc., 347(1995), 661-673.
  15. G. W. Johnson and D. L. Skoug, An Lp analytic Fourier-Feynman transform, Michigan Math. J., 26(1979), 103-127.
  16. G. W. Johnson and D. L. Skoug, Scale-invariant measurability in Wiener space, Pacific J. Math., 83(1979), 157-176.
  17. G. W. Johnson and D. L. Skoug, Notes on the Feynman integral, III: The Schroedinger equation, Pacific J. Math., 105(1983), 321-358.
  18. D. Skoug and D. Storvick, A survey of results involving transforms and convolutions in function space, Rocky Mountain J. Math., 34(2004), 1147-1175.
  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)