Acta Applicandae Mathematicae

, Volume 107, Issue 1–3, pp 25–48 | Cite as

Examples of Coorbit Spaces for Dual Pairs

Article

Abstract

In this paper we summarize and give examples of a generalization of the coorbit space theory initiated in the 1980’s by H.G. Feichtinger and K.H. Gröchenig. Coorbit theory has been a powerful tool in characterizing Banach spaces of distributions with the use of integrable representations of locally compact groups. Examples are a wavelet characterization of the Besov spaces and a characterization of some Bergman spaces by the discrete series representation of SL2(ℝ). We present examples of Banach spaces which could not be covered by the previous theory, and we also provide atomic decompositions for an example related to a non-integrable representation.

Keywords

Coorbit spaces Gelfand triples Representation theory of locally compact groups 

Mathematics Subject Classification (2000)

43A15 42B35 22D12 

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References

  1. 1.
    Chébli, H., Faraut, J.: Fonctions holomorphes à croissance modérée et vecteurs distributions. Math. Z. 248(3), 540–565 (2004) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Christensen, J., Ólafsson, G.: Coorbit spaces dual pairs. To be submitted Google Scholar
  3. 3.
    Duflo, M., Moore, C.C.: On the regular representation of a nonunimodular locally compact group. J. Funct. Anal. 21(2), 209–243 (1976) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Feichtinger, H.G., Gröchenig, K.H.: A unified approach to atomic decompositions via integrable group representations. In: Function Spaces and Applications, Lund, 1986. Lecture Notes in Math., vol. 1302, pp. 52–73. Springer, Berlin (1988) CrossRefGoogle Scholar
  5. 5.
    Feichtinger, H.G., Gröchenig, K.H.: Banach spaces related to integrable group representations and their atomic decompositions. I. J. Funct. Anal. 86(2), 307–340 (1989) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Feichtinger, H.G., Gröchenig, K.H.: Banach spaces related to integrable group representations and their atomic decompositions. II. Monatsh. Math. 108(2–3), 129–148 (1989) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Feichtinger, H.G., Gröchenig, K.H.: Irregular sampling theorems and series expansions of band-limited functions. J. Math. Anal. Appl. 167(2), 530–556 (1992) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Feichtinger, H.G., Pandey, S.S.: Recovery of band-limited functions on locally compact Abelian groups from irregular samples. Czechoslovak Math. J. 53(128)(2), 249–264 (2003) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gröchenig, K.H.: Describing functions: atomic decompositions versus frames. Monatsh. Math. 112(1), 1–42 (1991) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Harish-Chandra: Representations of semisimple Lie groups. IV. Am. J. Math. 77, 743–777 (1955) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Harish-Chandra: Representations of semisimple Lie groups. V. Am. J. Math. 78, 1–41 (1956) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Holschneider, M.: Wavelets. Oxford Mathematical Monographs. Clarendon Press/Oxford University Press, New York (1995). An analysis tool, Oxford Science Publications MATHGoogle Scholar
  13. 13.
    Knapp, A.W.: Representation Theory of Semisimple Groups. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (2001). An overview based on examples, Reprint of the 1986 original MATHGoogle Scholar
  14. 14.
    Krötz, B., Stanton, R.J.: Holomorphic extensions of representations. II. Geometry and harmonic analysis. Geom. Funct. Anal. 15(1), 190–245 (2005) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Ólafsson, G., Ørsted, B.: The holomorphic discrete series for affine symmetric spaces. I. J. Funct. Anal. 81(1), 126–159 (1988) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ólafsson, G., Schlichtkrull, H.: Representation theory, Radon transform and the heat equation on a Riemannian symmetric space. In: Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey. Contemp. Math., vol. 449, pp. 315–344. Am. Math. Soc., Providence (2008) Google Scholar
  17. 17.
    Pesenson, I.: A discrete Helgason-Fourier transform for Sobolev and Besov functions on noncompact symmtric spaces. In: Radon Transforms, Geometry, and Wavelets, vol. 464, pp. 231–247. Am. Math. Soc., Providence (2008) Google Scholar
  18. 18.
    Rauhut, H.: Time-Frequency and Wavelet Analysis of Functions with Symmetry Properties. Logos-Verlag (2005) Google Scholar
  19. 19.
    Triebel, H.: Function spaces on Lie groups, the Riemannian approach. J. Lond. Math. Soc. (2) 35(2), 327–338 (1987) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Triebel, H.: Characterizations of Besov-Hardy-Sobolev spaces: A unified approach. J. Approx. Theory 52(2), 162–203 (1988) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Warner, G.: Harmonic Analysis on Semi-Simple Lie Groups. I. Springer, New York (1972). Die Grundlehren der mathematischen Wissenschaften, Band 188 MATHGoogle Scholar
  22. 22.
    Wolf, J.A.: Harmonic Analysis on Commutative Spaces. Mathematical Surveys and Monographs, vol. 142. American Mathematical Society, Providence (2007) MATHGoogle Scholar
  23. 23.
    Zimmermann, G.: Coherent states from nonunitary representations. In: Topics in Multivariate Approximation and Interpolation, vol. 12, pp. 231–247. Elsevier, Amsterdam (2005) Google Scholar

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© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of mathematicsLouisiana State UniversityBaton RougeUSA

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