Skip to main content
SpringerLink
Log in

Interpolating polynomial wavelets on [−1,1]

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

The present paper gives a contribution of wavelet aspects to classical algebraic polynomial approximation theory. As is so often the case in classical approximation, the authors follow the pattern provided by the trigonometric polynomial case. Algebraic polynomial interpolating scaling functions and wavelets are constructed by using the interpolation properties of de la Vallée Poussin kernels with respect to the four kinds of Chebyshev weights. For the decomposition and reconstruction of a given function the structure of the involved matrices is studied in order to reduce the computational effort by means of fast discrete cosine and sine transforms.

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. D. Berthold, W. Hoppe and B. Silbermann, The numerical solution of the generalized airfoil equation, J. Integral Equations Appl. 4 (1992) 309–336.

    Google Scholar 

  2. M.R. Capobianco and W. Themistoclakis, On the boundedness of de la Vallée Poussin operators, East J. Approx. 7(4) (2001) 417–444.

    Google Scholar 

  3. C.K. Chui and E. Quak, Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory, eds. D. Braess and L.L. Schumaker (Birkhäuser, Basel, 1992) pp. 53–75.

    Google Scholar 

  4. A. Cohen, I. Daubechies and P. Vial, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmonic Anal. 1 (1993) 54–81.

    Google Scholar 

  5. I. Daubechies, Ten Lectures on Wavelets, CBMS Lecture Series, Vol. 61 (SIAM, Philadelphia, PA, 1992).

    Google Scholar 

  6. B. Fischer and J. Prestin, Wavelets based on orthogonal polynomials, Math. Comp. 66(220) (1997) 1593–1618.

    Google Scholar 

  7. B. Fischer and W. Themistoclakis, Orthogonal polynomial wavelets, Numer. Algorithms 30 (2002) 37–58.

    Google Scholar 

  8. T. Kilgore and J. Prestin, Polynomial wavelets on the interval, Constr. Approx. 12 (1996) 95–110.

    Google Scholar 

  9. T. Kilgore, J. Prestin and K. Selig, Polynomial wavelets and wavelet packet bases, Studia Sci. Math. Hungar. 33 (1997) 419–431.

    Google Scholar 

  10. T. Lyche, K. Mørken and E. Quak, Theory and algorithms for nonuniform spline wavelets, in: Multivariate Approximation Theory and Applications, eds. N. Dyn, D. Leviatan, D. Levin and A. Pinkus (Cambridge Univ. Press, Cambridge, 2001) pp. 152–187.

    Google Scholar 

  11. S. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. Amer. Math. Soc. 315(1) (1989) 69–87.

    Google Scholar 

  12. Y. Meyer, Ondelettes sur l’intervalle, Rev. Mat. Iberoamericana 7 (1992) 115–133.

    Google Scholar 

  13. C.A. Micchelli and Y. Xu, Using the matrix refinement equation for the construction of wavelets on invariant sets, Appl. Comput. Harmonic Anal. 1 (1994) 391–401.

    Google Scholar 

  14. P. Nevai, Mean convergence of Lagrange interpolation III, Trans. Amer. Math. Soc. 282 (1984) 669–698.

    Google Scholar 

  15. G. Plonka, K. Selig and M. Tasche, On the construction of wavelets on the interval, Adv. Comput. Math. 4 (1995) 357–388.

    Google Scholar 

  16. J. Prestin and K. Selig, Interpolating and orthonormal trigonometric wavelets, in: Signal and Image Representation in Combined Spaces, eds. J. Zeevi and R. Coifman (Academic Press, San Diego, 1998) pp. 201–255.

    Google Scholar 

  17. G. Steidl, Fast radix-p discrete cosine transform, Appl. Algebra Engrg. Comm. Comput. 3 (1992) 39–46.

    Google Scholar 

  18. G. Szegö, Orthogonal Polynomials, revised ed., AMS Colloquium Pubblications, Vol. XXIII (Amer. Math. Soc., New York, 1959).

    Google Scholar 

  19. M. Tasche, Polynomial wavelets on [−1,1], in: Approximation Theory, Wavelets and Applications, ed. S.P. Singh (Kluwer, Dordrecht, 1995) pp. 49–70.

    Google Scholar 

  20. W. Themistoclakis, Trigonometric wavelet interpolation in Besov spaces, Facta Univ. (Nis) Ser. Math. Inform. 14 (1999) 49–70.

    Google Scholar 

  21. W. Themistoclakis, Some interpolating operators of de la Vallée Poussin type, Acta Math. Hungar. 84(3) (1999) 221–235.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. C.N.R., Istituto per le Applicazioni del Calcolo “M. Picone”, Sezione di Napoli Via P. Castellino, 111, 80131, Napoli, Italy

    M. R. Capobianco & W. Themistoclakis

Authors
  1. M. R. Capobianco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. Themistoclakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. R. Capobianco.

Additional information

Communicated by R.-Q. Jia

Dedicated to Prof. Guiseppe Mastroianni on the occasion of his 65th birthday.

AMS subject classification

65D05, 65T60

Rights and permissions

Reprints and permissions

About this article

Cite this article

Capobianco, M.R., Themistoclakis, W. Interpolating polynomial wavelets on [−1,1]. Adv Comput Math 23, 353–374 (2004). https://doi.org/10.1007/s10444-004-1828-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-004-1828-2

Keywords

Search

Navigation