The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

A REPRESENTATION FOR AN INVERSE GENERALIZED FOURIER-FEYNMAN TRANSFORM ASSOCIATED WITH GAUSSIAN PROCESS ON FUNCTION SPACE

  • Choi, Jae Gil (School of General Education, Dankook University)
  • Received : 2020.08.07
  • Accepted : 2021.08.27
  • Published : 2021.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we suggest a representation for an inverse transform of the generalized Fourier-Feynman transform on the function space Ca,b[0, T]. The function space Ca,b[0, T] is induced by the generalized Brownian motion process with mean function a(t) and variance function b(t). To do this, we study the generalized Fourier-Feynman transform associated with the Gaussian process Ƶk of exponential-type functionals. We then establish that a composition of the Ƶk-generalized Fourier-Feynman transforms 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. Ph.D. Thesis, University of Minnesota, Minneapolis, 1972.
  2. R.H. Cameron: Some examples of Fourier-Wiener transforms of analytic functionals. Duke Math. J. 12 (1945), 485-488. https://doi.org/10.1215/S0012-7094-45-01243-9
  3. R.H. Cameron & W.T. Martin: Fourier-Wiener transforms of analytic functionals. Duke Math. J. 12 (1945), 489-507. https://doi.org/10.1215/S0012-7094-45-01244-0
  4. R.H. Cameron & W.T. Martin: Fourier-Wiener transforms of functionals belonging to L2 over the space C. Duke Math. J. 14 (1947), 99-107. https://doi.org/10.1215/S0012-7094-47-01409-9
  5. R.H. Cameron & D.A. Storvick: An L2 analytic Fourier-Feynman transform. Michigan Math. J. 23 (1976), 1-30.
  6. S.J. Chang & J.G. Choi: Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral. Arch. Math. 106 (2016), 591-600. https://doi.org/10.1007/s00013-016-0899-x
  7. S.J. Chang & J.G. Choi: A representation for the inverse generalised Fourier-Feynman transform via convolution product on function space. Bull. Aust. Math. Soc. 95 (2017), 424-435. https://doi.org/10.1017/S0004972716001039
  8. S.J. Chang, J.G. Choi & A.Y. Ko: Multiple generalized analytic Fourier-Feynman transform via rotation of Gaussian paths on function space. Banach J. Math. Anal. 9 (2015), 58-80. https://doi.org/10.15352/bjma/09-4-4
  9. S.J. Chang, J.G. Choi & D. Skoug: Integration by parts formulas involving generalized Fourier-Feynman transforms on function space. Trans. Amer. Math. Soc. 355 (2003), 2925-2948. https://doi.org/10.1090/S0002-9947-03-03256-2
  10. S.J. Chang, H.S. Chung & D. Skoug: Integral transforms of functionals in L2 (Ca,b[0, T]). J. Fourier Anal. Appl. 15 (2009), 441-462. https://doi.org/10.1007/s00041-009-9076-y
  11. S.J. Chang, H.S. Chung & D. Skoug: Convolution products, integral transforms and inverse integral transforms of functionals in L2(C0[0, T]). Integral Transform Spec. Funct. 21 (2010), 143-151. https://doi.org/10.1080/10652460903063382
  12. S.J. Chang, H.S. Chung & D. Skoug: Some basic relationships among transforms, convolution products, first variations and inverse transforms. Cent. Eur. J. Math. 11 (2013), 538-551.
  13. S.J. Chang, W.G. Lee & J.G. Choi: L2-sequential transforms on function space. J. Math. Anal. Appl. 421 (2015), 625-642. https://doi.org/10.1016/j.jmaa.2014.07.034
  14. S.J. Chang & D. Skoug: Generalized Fourier-Feynman transforms and a first variation on function space. Integral Transforms Spec. Funct. 14 (2003), 375-393. https://doi.org/10.1080/1065246031000074425
  15. J.G. Choi & S.J. Chang: Generalized Fourier-Feynman transform and sequential transforms on function space. J. Korean Math. Soc. 49 (2012), 1065-1082. https://doi.org/10.4134/JKMS.2012.49.5.1065
  16. J.G. Choi & D. Skoug: Further results involving the Hilbert space L2a,b[0, T]. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 27 (2020), 1-11.
  17. G.W. Johnson & D.L. Skoug: An Lp analytic Fourier-Feynman transform. Michigan Math. J. 26 (1979), 103-127.
  18. B.S. Kim & D.L. Skoug: Integral transforms of functionals in L2(C0[0, T]). Rocky Mountain J. Math. 33 (2003), 1379-1393. https://doi.org/10.1216/rmjm/1181075469
  19. Y.-J. Lee: Integral transforms of analytic functions on abstract Wiener spaces. J. Funct. Anal. 47 (1982), 153-164. https://doi.org/10.1016/0022-1236(82)90103-3
  20. Y.-J. Lee: Unitary operators on the space of L2-functions over abstract Wiener spaces. Soochow J. Math. 13 (1987), 165-174.
  21. D. Skoug & D. Storvick: A survey of results involving transforms and convolutions in function space. Rocky Mountain J. Math. 34 (2004), 1147-1175. https://doi.org/10.1216/rmjm/1181069848
  22. 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. https://doi.org/10.1215/ijm/1256052816
  23. J. Yeh: Stochastic Processes and the Wiener Integral. Marcel Dekker Inc, New York, 1973.