DOI QR코드

DOI QR Code

GENERALIZED FOURIER-WIENER FUNCTION SPACE TRANSFORMS

  • 발행 : 2009.03.31

초록

In this paper, we define generalized Fourier-Hermite functionals on a function space $C_{a,b}[0,\;T]$ to obtain a complete orthonormal set in $L_2(C_{a,b}[0,\;T])$ where $C_{a,b}[0,\;T]$ is a very general function space. We then proceed to give a necessary and sufficient condition that a functional F in $L_2(C_{a,b}[0,\;T])$ has a generalized Fourier-Wiener function space transform ${\cal{F}}_{\sqrt{2},i}(F)$ also belonging to $L_2(C_{a,b}[0,\;T])$.

키워드

참고문헌

  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 and 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 and W. T. Martin, Fourier-Wiener transforms of functionals belonging to $L_2$ over the space Ć, Duke Math. J. 14 (1947), 99–107. https://doi.org/10.1215/S0012-7094-47-01409-9
  4. R. H. Cameron and W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Ann. of Math. 48 (1947), 385–392. https://doi.org/10.2307/1969178
  5. 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
  6. K. S. Chang, B. S. Kim, and I. Yoo, Integral transform and convolution of analytic functionals on abstract Wiener space, Numer. Funct. Anal. Optim. 21 (2000), no. 1-2, 97–105. https://doi.org/10.1080/01630560008816942
  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, 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
  9. S. J. Chang and D. L. 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
  10. B. S. Kim and D. Skoug, Integral transforms of functionals in $L_{2}$($C_{0}$[0, T]), Rocky Mountain J. Math. 33 (2003), no. 4, 1379–1393. https://doi.org/10.1216/rmjm/1181075469
  11. Y. J. Lee, Integral transforms of analytic functions on abstract Wiener spaces, J. Funct. Anal. 47 (1982), no. 2, 153–164. https://doi.org/10.1016/0022-1236(82)90103-3
  12. Y. J. Lee, Unitary operators on the space of $L^2$-functions over abstract Wiener spaces, Soochow J. Math. 13 (1987), no. 2, 165–174.
  13. 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
  14. E. Nelson, Dynamical Theories of Brownian Motion (2nd edition), Math. Notes, Princeton University Press, Princeton, 1967.
  15. J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973.
  16. 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.

피인용 문헌

  1. Relationships Involving Transforms and Convolutions Via the Translation Theorem vol.32, pp.2, 2014, https://doi.org/10.1080/07362994.2013.877350
  2. SERIES EXPANSIONS OF THE ANALYTIC FEYNMAN INTEGRAL FOR THE FOURIER-TYPE FUNCTIONAL vol.19, pp.2, 2012, https://doi.org/10.7468/jksmeb.2012.19.2.87
  3. REFLECTION PRINCIPLES FOR GENERAL WIENER FUNCTION SPACES vol.50, pp.3, 2013, https://doi.org/10.4134/JKMS.2013.50.3.607
  4. Some basic relationships among transforms, convolution products, first variations and inverse transforms vol.11, pp.3, 2013, https://doi.org/10.2478/s11533-012-0148-x
  5. Self-adjoint oscillator operator from a modified factorization vol.375, pp.22, 2011, https://doi.org/10.1016/j.physleta.2011.04.012
  6. Generalized conditional transform with respect to the Gaussian process on function space vol.26, pp.12, 2015, https://doi.org/10.1080/10652469.2015.1070149
  7. generalized analytic Fourier–Feynman transform vol.29, pp.9, 2018, https://doi.org/10.1080/10652469.2018.1497024