Advertisement

Proceedings - Mathematical Sciences

, Volume 120, Issue 2, pp 169–183 | Cite as

Segal-Bargmann transform and Paley-Wiener theorems on M(2)

  • E. K. NarayananEmail author
  • Suparna Sen
Article
  • 72 Downloads

Abstract

We study the Segal-Bargmann transform on M(2). The range of this transform is characterized as a weighted Bergman space. In a similar fashion Poisson integrals are investigated. Using a Gutzmer’s type formula we characterize the range as a class of functions extending holomorphically to an appropriate domain in the complexification of M(2). We also prove a Paley-Wiener theorem for the inverse Fourier transform.

Keywords

Segal-Bargmann transform Poisson integrals Paley-Wiener theorem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Goodman RW, Analytic and entire vectors for representations of Lie groups, Trans. Am. Math. Soc. 143 (1969) 55–76zbMATHGoogle Scholar
  2. [2]
    Goodman R W, Complex Fourier analysis on a nilpotent Lie group, Trans. Am. Math. Soc. 160 (1971) 373–391zbMATHGoogle Scholar
  3. [3]
    Hall B C, The Segal-Bargmann ‘coherent state’ transform for compact Lie groups, J. Funct. Anal. 122(1) (1994) 103–151zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Hall B C and Lewkeeratiyutkul W, Holomorphic Sobolev spaces and the generalized Segal-Bargmann transform, J. Funct. Anal. 217(1) (2004) 192–220CrossRefMathSciNetGoogle Scholar
  5. [5]
    Hall B C and Mitchell J J, The Segal-Bargmann transform for non compact symmetric spaces of the complex type, J. Funct. Anal. 227(2) (2005) 338–371zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Krötz B, Ólafsson and Stanton R J, The image of the heat kernel transform on Riemannian symmetric spaces of the non compact type, Int. Math. Res. Not. 22 (2005) 1307–1329CrossRefGoogle Scholar
  7. [7]
    Krötz B, Thangavelu S and Xu Y, The heat kernel transform for the Heisenberg group, J. Funct. Anal. 225(2) (2005) 301–336CrossRefGoogle Scholar
  8. [8]
    Pasquale A, A Paley-Wiener theorem for the inverse spherical transform, Pacific J. Math. 193(1) (2000) 143–176zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Stenzel M B, The Segal-Bargmann transform on a symmetric space of compact type, J. Funct. Anal. 165(1) (1999) 44–58zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Sugiura M, Fourier series of smooth functions on compact Lie groups, Osaka J. Math. 8 (1971) 33–47zbMATHMathSciNetGoogle Scholar
  11. [11]
    Thangavelu S, A Paley-Wiener theorem for the inverse Fourier transform on some homogeneous spaces, Hiroshima Math. J. 37(2) (2007) 145–159zbMATHMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 2010

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

Personalised recommendations