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Wavelet Transform of Operators and Functional Calculus

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Complex Methods for Partial Differential Equations

Part of the book series: International Society for Analysis, Applications and Computation ((ISAA,volume 6))

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Abstract

We describe a construction of wavelets (coherent states) in Banach spaces generated by “admissible” group representations. Our main examples are operator valued Segal-Bargmann type spaces and corresponding functional calculi.

Supported by Grant INTAS 93-10322.

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References

  1. Ali, S.T., Antoine, J.-P., Gazeau, J.-P., Mueller, U.: Coherent states and their generalizations: A mathematical overview. Rev. Math. Phys. 7 (1995), 1013–1104.

    Article  MathSciNet  MATH  Google Scholar 

  2. Anderson, R.F.V.: The Weyl functional calculus. J. Funct. Anal. 4 (1969), 240–267.

    Article  MATH  Google Scholar 

  3. Arveson, W.: The harmonic analysis of automorphisms groups. Amer. Math. Soc. Summer Institute. Amer. Math. Soc., Providence, R.I.,1980.

    Google Scholar 

  4. Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform, I. Comm. Pure Appl. Math. 3 (1961), 215–228.

    MathSciNet  Google Scholar 

  5. Berezin, F.A.: Method of second quantization. Nauka, Moscow, 1988.

    Google Scholar 

  6. Cnops, J., Kisil, V.V.: Monogenic functions and representations of nilpotent Lie groups in quantum mechanics. Math. Meth. Appl. Sci. 22 (1998), 353–373. http://xxx.lanl.gov/abs/math/9806150/.

    Article  MathSciNet  Google Scholar 

  7. Coburn, L.A.: Berezin-Toeplitz quantization. Algebraic methods in operator theory. Birkhäuser Verlag, New York, 1994, 101–108.

    Book  Google Scholar 

  8. Connes, A.: Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. (4) 6 (1973), 133–252.

    MathSciNet  MATH  Google Scholar 

  9. Daletski, Y.: On representation of solutions of operator equations in a form of functional integrals. Dokl. Akad. Nauk SSSR 134 (1960), 1013–1016.

    MathSciNet  Google Scholar 

  10. Daubechies, I.: Ten lectures on wavelets, CBMS-NSF regional conference series in appl. math. 61. SIAM, Philadelphia, PA, 1992.

    Google Scholar 

  11. Folland, G.B.: Harmonic analysis in phase space. Princeton Univ. Press, Princeton, N.J., 1989.

    Google Scholar 

  12. Heil, C.E., Walnut, D.F.: Continuous and discrete wavelet transforms. SIAM Rev. 31 (1989), 628–666.

    Article  MathSciNet  MATH  Google Scholar 

  13. Howe, R.: On the role of the Heisenberg group in harmonic analysis. Bull. Amer. Math. Soc. (N.S.) 3 (1980), 821–843.

    Article  MathSciNet  MATH  Google Scholar 

  14. Kisil, V.V.: Möbius transformations and monogenic functional calculus. Electron. Res. Announc. Amer. Math. Soc. 2 (1) (1996), 26–33. http://www.ams.org

    Article  MathSciNet  Google Scholar 

  15. Kisil, V.V. : Two approaches to non-commutative geometry. Complex methods for partial differential equations. Eds. H. Begehr et al. Kluwer, Dordrecht, 1999, 217–246. funct-an/9703001http://xxx.lanl.gov/abs/funct-an/9703001/.

    Google Scholar 

  16. Kisil, V.V. : Analysis in ℝ1,1 or the principal function theory. Complex Variables, Theory Appl., (to appear). http://xxx.lanl.gov/abs/funct-an/9712003/.

  17. Kisil, V.V.: Wavelets in Banach spaces. Preprint, 1998. http://xxx.lanl.gov/abs/math/9807141/.

    Google Scholar 

  18. Krein, M.G. : Selected Works, I. With a biographical sketch of Krein by D.Z. Arov et al. Akad. Nauk Ukrainy Inst. Mat., Kiev, 1993 (Russian).

    Google Scholar 

  19. Reed, M., Simon, B.: Functional analysis. 2nd ed., Academic Press, Orlando, 1980.

    MATH  Google Scholar 

  20. Segal, I.E.: Mathematical problems of relativistic physics, Proceedings of the Summer Seminar, Boulder, Colorado, 1960, II. Amer. Math. Soc., Providence, R.I., 1963.

    Google Scholar 

  21. Takesaki, M.: Structure of factors and automorphism groups. Amer. Math. Soc., Providence, R.I., 1983.

    Google Scholar 

  22. Taylor, M. E.: Noncommutative harmonic analysis. Amer. Math. Soc., Providence, R.I., 1986.

    Google Scholar 

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© 1999 Kluwer Academic Publishers

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Kisil, V.V. (1999). Wavelet Transform of Operators and Functional Calculus. In: Begehr, H.G.W., Celebi, A.O., Tutschke, W. (eds) Complex Methods for Partial Differential Equations. International Society for Analysis, Applications and Computation, vol 6. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3291-6_21

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  • DOI: https://doi.org/10.1007/978-1-4613-3291-6_21

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-3293-0

  • Online ISBN: 978-1-4613-3291-6

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