Skip to main content
Log in

Riesz bounds of Wilson bases generated byB-splines

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

In this paper, we are concerned with biorthogonal Wilson bases having B-splines as well as powers of sinc functions as window functions. We prove properties of B-splines and exponential Euler splines and use these properties to estimate the Riesz bounds of the Wilson bases.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Auscher, P., Weiss, G., and Wickerhauser, M.V. (1992). Local sine and cosine bases of Coifman and Meyer and the construction of smooth wavelets, in:Wavelets: A Tutorial in Theory and Applications, Chui, C.K., Ed., Academic Press, Boston, 237–256.

    Google Scholar 

  2. Benedetto, J.J. and Walnut, D.F. (1994). Gabor frames ofL 2 and related spaces, in:Wavelets: Mathematics and Applications, Benedetto, J.J. and Frazier, M.W., Eds., CRC Press, Boca Raton, FL., 97–162.

    Google Scholar 

  3. Bittner, K., Chui, C.K., and Prestin, J. (1997). Multivariate cosine wavelets, in:Multivariate Approximation and Splines, Nürnberger, G., Schmidt, J.W., and Walz, G., Eds., Birkhäuser, Basel, 29–45.

    Google Scholar 

  4. Bittner, K. Private communication.

  5. Bittner, K. Ph.D. thesis, TU München, 1999.

  6. Bölcskei, H., Gröchenig, K., Hlawatsch, F., and Feichtinger, H.G. (1997). Oversampled Wilson systems,IEEE Signal Proc. Lett.,4 (4), 106–108.

    Article  Google Scholar 

  7. Chui, C.K. and Shi, X. (1997). Characterization of biorthogonal cosine wavelets.J. Fourier Anal. Appl.,3, 559–575.

    MATH  MathSciNet  Google Scholar 

  8. Chui, C.K. and Shi, X. (1997). A study of biorthogonal sinusoidal wavelets, in:Surface Fitting and Multiresolution Methods, Le Méhauté, A., Rabut, C., and Schumaker, L.L., Eds., Vanderbuilt University Press, Nashville, TN, 51–66.

    Google Scholar 

  9. Coifman, R.R. and Meyer, Y. (1991). Remarques sur l’analyse de Fourier a fenêtre,C.R. Acad. Sci. Paris,312, 259–261.

    MATH  MathSciNet  Google Scholar 

  10. Coifman, R.R. and Meyer, Y. (1995). Gaussian bases,Appl. Comput. Harmon. Anal.,2, 299–302.

    Article  MATH  MathSciNet  Google Scholar 

  11. Daubechies, I. (1992). Ten Lectures on Wavelets,SIAM, Philadelphia.

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Feichtinger, H.G. and Strohmer, T., Eds., (1998).Gabor Analysis and Algorithms: Theory and Applications, Birkhäuser, Basel.

    Google Scholar 

  14. Hernandez, E. and Weiss, G. (1996).A First Course on Wavelets, CRC Press, Boca Raton, FL.

    MATH  Google Scholar 

  15. Malvar, H. (1990). Lapped transforms for efficient transform/subband coding,IEEE Trans. Acoust. Speech Sig. Proc.,38/6, 969–978.

    Article  Google Scholar 

  16. Plonka, G. and Tasche, M. (1993). Periodic spline wavelets.Appl. Comput. Harmon. Anal.,2, 1–14.

    Article  MathSciNet  Google Scholar 

  17. Marsden, J.E. and Hoffmann, M.J. (1987).Basic Complex Analysis, W.H. Freeman and Company, New York.

    MATH  Google Scholar 

  18. Schoenberg, I.J. (1972). Cardinal interpolation and spline functions IV: The exponential Euler spline, in:Linear Operators and Approximation, Butzer, P.J., Kahane, J.P., and Sz.-Nagy, B., Eds. Birkhäuser, 382–404.

  19. Wilson, K. (1987). Generalized Wannier functions, preprint.

  20. Jetter, K., Riemenschneider, S.D., and Sivakumar, N. (1991). Schoenberg’s exponential Euler spline curves.Proc. Royal Soc. Edinburgh,118A, 21–35.

    MathSciNet  Google Scholar 

  21. Yaglom, A.M. and Yaglom, I.M. (1964).Challenging Problems with Elementary Solutions I, Holden-Day Inc., San Francisco, CA.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by A.J.E.M. Janssen

Rights and permissions

Reprints and permissions

About this article

Cite this article

Trebels, B., Steidl, G. Riesz bounds of Wilson bases generated byB-splines. The Journal of Fourier Analysis and Applications 6, 171–184 (2000). https://doi.org/10.1007/BF02510659

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02510659

Math Subject Classifications

Keywords and Phrases

Navigation