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Integral Transforms of Functionals in L 2(C a,b [0,T])

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Abstract

In this paper we use generalized Fourier-Hermite functionals 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 an integral transform ℱ γ,β F also belonging to L 2(C a,b [0,T]).

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References

  1. Cameron, R.H.: Some examples of Fourier-Wiener transforms of analytic functionals. Duke Math. J. 12, 485–488 (1945)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cameron, R.H., Martin, W.T.: Fourier-Wiener transforms of analytic functionals. Duke Math. J. 12, 489–507 (1945)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cameron, R.H., Martin, W.T.: Fourier-Wiener transforms of analytic functionals belonging to L 2 over the space C. Duke Math. J. 14, 99–107 (1947)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cameron, R.H., Martin, W.T.: The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. Ann. Math. 48, 385–392 (1947)

    Article  MathSciNet  Google Scholar 

  5. Cameron, R.H., Storvick, D.A.: An L 2 analytic Fourier-Feynman transform. Mich. Math. J. 23, 1–30 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chang, S.J., Chung, D.M.: Conditional function space integrals with applications. Rocky Mt. J. Math. 26, 37–62 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chang, S.J., Skoug, D.L.: Generalized Fourier-Feynman transforms and a first variation on function space. Integral Transforms Spec. Funct. 14, 375–393 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chang, K.S., Kim, B.S., Yoo, I.: Integral transforms and convolution of analytic functionals on abstract Wiener spaces. Numer. Funct. Anal. Optim. 21, 97–105 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chang, S.J., Choi, J.G., Skoug, D.: Integration by parts formulas involving generalized Fourier-Feynman transforms on function space. Trans. Am. Math. Soc. 355, 2925–2948 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Keener, J.P.: Principles of Applied Mathematics. Addison–Wesley, Redwood City (1988)

    MATH  Google Scholar 

  11. Kim, B.S., Skoug, D.: Integral transforms of functionals in L 2(C 0[0,T]). Rocky Mt. J. Math. 33, 1379–1393 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lee, Y.J.: Integral transforms of analytic functions on abstract Wiener spaces. J. Funct. Anal. 47, 153–164 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lee, Y.J.: Unitary operators on the space of L 2-functions over abstract Wiener spaces. Soochow J. Math. 13, 165–174 (1987)

    MATH  MathSciNet  Google Scholar 

  14. Nelson, E.: Dynamical Theories of Brownian Motion, 2nd edn. Math. Notes. Princeton University Press, Princeton (1967)

    MATH  Google Scholar 

  15. Paley, R.E.A.C., Wiener, N.: Fourier Transforms in the Complex Domain. Amer. Math. Colloq. Publ., vol. 19. Am. Math. Soc., Providence (1934)

    MATH  Google Scholar 

  16. Paley, R.E.A.C., Wiener, N., Zygmund, A.: Notes on random functions. Math. Z. 37, 647–668 (1933)

    Article  MathSciNet  Google Scholar 

  17. Skoug, D., Storvick, D.: A survey of results involving transforms and convolution in function space. Rocky Mt. J. Math. 34, 1147–1175 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Yeh, J.: Singularity of Gaussian measures on function spaces induced by Brownian motion processes with non-stationary increments. Ill. J. Math. 15, 37–46 (1971)

    MATH  Google Scholar 

  19. Yeh, J.: Stochastic Processes and the Wiener Integral. Dekker, New York (1973)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Dankook University, Cheonan, 330-714, South Korea

    Seung Jun Chang & Hyun Soo Chung

  2. Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588-0130, USA

    David Skoug

Authors
  1. Seung Jun Chang
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  2. Hyun Soo Chung
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  3. David Skoug
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Correspondence to Seung Jun Chang.

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Communicated by Christian Houdré.

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Chang, S.J., Chung, H.S. & Skoug, D. Integral Transforms of Functionals in L 2(C a,b [0,T]). J Fourier Anal Appl 15, 441–462 (2009). https://doi.org/10.1007/s00041-009-9076-y

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  • DOI: https://doi.org/10.1007/s00041-009-9076-y

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