INTEGRAL TRANSFORMS AND INVERSE INTEGRAL TRANSFORMS WITH RELATED TOPICS ON FUNCTION SPACE I

  • Published : 2009.11.30

Abstract

In this paper we establish various relationships among the generalized integral transform, the generalized convolution product and the first variation for functionals in a Banach algebra S($L_{a,b}^2$[0, T]) introduced by Chang and Skoug in [14]. We then derive an inverse integral transform and obtain several relationships involving inverse integral transforms.

Keywords

References

  1. 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
  2. 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
  3. R.H. Cameron & W.T. Martin: Fourier-Wiener transforms of functionals belonging to $L_2$ over the space C. Duke Math. J. 14 (1947), 99-107. https://doi.org/10.1215/S0012-7094-47-01409-9
  4. R.H. Cameron & D.A. Storvick: An $L_2$ analytic Fourier-Feynman transform. Michigan Math. J. 23 (1976), 1-30. https://doi.org/10.1307/mmj/1029001617
  5. R.H. Cameron & D.A. Storvick: Some Banach algebras of analytic Feynman integrable functionals, in Analytic Functions. Springer Lecture Notes in Mathematics, 798, Berlin, 1980, 18-67.
  6. S.J. Chang, J.G. Choi & H.S. Chung: Transforms and convolutions of functionals in a Fresnel type class. submitted.
  7. S.J. Chang, J.G. Choi & S.D. Lee: Fresnel type class on function space. J. Math. Soc. Math. Educ. Ser. B Pure Appl. Math. 16 (2009), 107-119.
  8. 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
  9. S.J. Chang & D.M. Chung: Conditional function space integrals with applications. Rocky Mountain J. Math. 26 (1996), 37-62. https://doi.org/10.1216/rmjm/1181072102
  10. S.J. Chang & H.S. Chung: Generalized analytic Feynman integrals involving generalized analytic Fourier-Feynman transforms and generalized integral transforms. J. Chungcheong Math. Soc. 21 (2008), 231-246.
  11. S.J. Chang & H.S. Chung: Generalized Fourier-Wiener function space transforms. J. Korean Math. Soc. 46 (2009), no. 2, 327-345. https://doi.org/10.4134/JKMS.2009.46.2.327
  12. S.J. Chang, H.S. Chung & D. Skoug: Integral transforms of functionals in $L^2(C_{a,b}[0,T])$ to appear in J. Fourier Anal. Appl.
  13. K.S. Chang, B.S. Kim & I. Yoo: Integral transform and convolution of analytic functionals on abstract Wiener spaces. Numer. Funct. Anal. Optim. 21 (2000), 97-105. https://doi.org/10.1080/01630560008816942
  14. S.J. Chang & D. Skoug: Generalized Fourier-Feynman transforms and a first variation on function space. Integral Transforms and Special Functions 14 (2003), 375-393. https://doi.org/10.1080/1065246031000074425
  15. G.W. Johnson & M.L. Lapidus: The Feynman Integral and Feynman’s Operational Calculus, Oxford Mathematical Monographs, The Clarendon Press, Oxford, 2000.
  16. G.W. Johnson & D.L. Skoug: Notes on the Feynman integral, I. Pacific J. Math. 93 (1981), 313-324. https://doi.org/10.2140/pjm.1981.93.313
  17. B.J. Kim, B.S. Kim & D. Skoug: Integral transforms, convolution products, and first variations. Int. J. Math. Math. Soc. 11 (2004), 579-598.
  18. B.S. Kim & D. Skoug: Integral transforms of functionals in $L_2(C_0[0,T])$. 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 $L^2-functions$ over abstract Wiener spaces. Soochow. J. Math. 13 (1987),165-174.
  21. E. Nelson: Dynamical Theories of Brownian Motion(2nd edition). Math. Notes, Priceton University Press, Princeton, 1967.
  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.
  23. J. Yeh: Stochastic Processes and the Wiener Integral. Marcel Dekker, Inc., New York, 1973.