Skip to main content
SpringerLink
Log in

A Riesz basis for Bargmann-Fock space related to sampling and interpolation

  • Published:
Arkiv för Matematik

Abstract

It is shown that the Bargmann-Fock spaces of entire functions, Ap (C),p≧1 have a bounded unconditional basis of Wilson type [DJJ] which is closely related to the reproducing kernel. From this is derived a new sampling and interpolation result for these spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Auslander, L. andTolimieri, R., Radar Ambiguity functions and group theory,SIAM J. Math. Anal. 1 (1979), 847–897.

    MATH  MathSciNet  Google Scholar 

  2. Bargmann, V., On a Hilbert space of analytic functions and an associated integral transform I.,Comm. Pure Appl. Math. 14 (1961), 187–214.

    Article  MATH  MathSciNet  Google Scholar 

  3. Daubechies, I., The wavelet transform, time-frequency localization and signal analysis,IEEE Trans. Inform. Theory 36 (1990), 961–1005.

    Article  MATH  MathSciNet  Google Scholar 

  4. Daubechies, I. andGrossman, A., Frames in the Bargmann space of entire functions,Comm. Pure Appl. Math. 41 (1985), 151–164.

    Article  Google Scholar 

  5. Daubechies, I., Grossmann, A. andMeyer, Y., Painless nonorthogonal expansions,J. Math. Phys. 27 (1986), 1271–1283.

    Article  MATH  MathSciNet  Google Scholar 

  6. Daubechies, I., Jaffard, S. andJourné, J.-L., A simple Wilson basis with exponential decay,SIAM J. Math. Anal.,22 (1991), 554–572.

    Article  MATH  MathSciNet  Google Scholar 

  7. Duffin, R. J. andSchaeffer, A. C., A class of nonharmonic Fourier series,Trans. Amer. Math. Soc. 72 (1952), 341–366.

    Article  MATH  MathSciNet  Google Scholar 

  8. Feichtinger, H., Atomic decompositions of modulation spaces,Rocky Mountain J. Math. 19 (1989), 113–125.

    Article  MATH  MathSciNet  Google Scholar 

  9. Feichtinger, H. andGröchenig, K., A unified approach to atomic decompositions through integrable group representations,Function Spaces and Applications, M. Cwikel et al., eds., Lecture Notes in Math. 1302, Springer-Verlag, Berlin, New York, 1988, pp. 52–73.

    Chapter  Google Scholar 

  10. Feichtinger, H. andGröchenig, K., Banach spaces related to integrable group representations and their atomic decompositions, I,J. Funct. Anal. 86 (1989), 307–340.

    Article  MATH  MathSciNet  Google Scholar 

  11. Feichtinger, H. andGröchenig, K., Banach spaces related to integrable group representations and their atomic decompositions, II,Monatsh. f. Math. 108 (1989), 129–148.

    Article  MATH  Google Scholar 

  12. Feichtinger, H., Gröchenig, K. andWalnut, D., Wilson Bases in Modulation Spaces,Math. Nachr. 155 (1992), 7–17.

    Article  MATH  MathSciNet  Google Scholar 

  13. Folland, G., Harmonic Analysis on Phase Space,Annals of Math. Studies No. 122, Princeton Univ. Press, Princeton, 1989.

    Google Scholar 

  14. Gabor, D., Theory of communications,J. Inst. Elec. Eng. (London)93 (1946), 429–457.

    Google Scholar 

  15. Gröchenig, K., Describing functions: Atomic decompositions vs. frames,Monatsh. Math. 112 (1991), 1–41.

    Article  MATH  MathSciNet  Google Scholar 

  16. Heil, C. andWalnut, D., Continuous and discrete wavelet transforms,SIAM Review 31 (1989), 628–666.

    Article  MATH  MathSciNet  Google Scholar 

  17. Janson, S., Peetre, J. andRochberg, R., Hankel forms on the Fock space,Rev. Mat. Iberoam. 3 (1987), 61–138.

    MATH  MathSciNet  Google Scholar 

  18. Janssen, A. J. E. M., Bargmann transform, Zak transform, and coherent states,J. Math. Phys. 23 (1982), 720–731.

    Article  MATH  MathSciNet  Google Scholar 

  19. Janssen, A. J. E. M., The Zak transform: a signal transform for sampled time-continuous signals,Philips J. Res. 43 (1988), 23–69.

    MATH  MathSciNet  Google Scholar 

  20. Janssen, A. J. E. M., Gabor representations of generalized functions,J. Math. Appl. 80 (1981), 377–394.

    Google Scholar 

  21. Rainville, E.,Special Functions, Chelsea Pub. Co., New York, 1960.

    MATH  Google Scholar 

  22. Seip, K. andWallstén, R., Sampling and interpolation in the Bargmann-Fock space, preprint, Institut Mittag-Leffler.

  23. Zak, J., Finite translations in solid state physics,Phys. Rev. Lett. 19 (1967), 1385–1397.

    Article  Google Scholar 

  24. Zygmund, A.,Trigonometric Series, Vol. I, Cambridge Univ. Press, 1959.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, 06269-3009, Storrs, CT

    K. Gröchenig

  2. Department of Mathematical Sciences, George Mason University, Fairfax, VA

    D. Walnut

Authors
  1. K. Gröchenig
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Walnut
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Partially supported by grant AFOSR 90-0311.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gröchenig, K., Walnut, D. A Riesz basis for Bargmann-Fock space related to sampling and interpolation. Ark. Mat. 30, 283–295 (1992). https://doi.org/10.1007/BF02384875

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02384875

Keywords

Search

Navigation