Analysis in Theory and Applications

, Volume 22, Issue 4, pp 301–318 | Cite as

Generalization of the interaction between Haar approximation and polynomial operators to higher order methods

  • François Chaplais


In applications it is useful to compute the local average of a function f(u) of an input u from empirical statistics on u. A very simple relation exists when the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.

Key words

Strang and Fix conditions product approximation Hermite interpolation wavelets 

AMS(2000) subject classification

41A05 41A35 42C40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Sanders, J. and Verhulst, F., Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York, 1985.zbMATHGoogle Scholar
  2. [2]
    Daubechies, I., Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992.zbMATHGoogle Scholar
  3. [3]
    Meyer, Y., Wavelets, and Operators, Advanced Mathematics, Cambridge University Press, 1992.Google Scholar
  4. [4]
    Mallat, S., A Wavelet Tour of Signal Processing, Second ed., Academic Press, 1999.Google Scholar
  5. [5]
    Strang, G. and Fix, G., A Fourier Analysis of the Finite Element Variational Method, Construct. Aspects of Funct. Anal., 1971, 796–830.Google Scholar
  6. [6]
    Daubechies, I., Orthonormal Bases of Compactly Supported Wavelets, Commun. on Pure and Appl. Math., 41(1988), 909–996.zbMATHMathSciNetGoogle Scholar
  7. [7]
    Beylkin, G., On the Fast Algorithm for Multiplication of Functions in the Wavelet Bases, in: Proceedings of the International Conference “Wavelets and Applications”, Toulouse, 1992.Google Scholar
  8. [8]
    Deslauriers, G and Dubuc, S., Symmetric Iterative Interpolation, Constr. Approx., 5(1989), 49–68.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Donoho, D., Interpolating Wavelet Transforms, Tech. Rep., Department of Statistics, Stanford University, 1992.Google Scholar
  10. [10]
    Jaffard, S., Exposants de Hölder en des Points Donnés et Coefficients D’ondelettes, C. R. Acad. Sci. Paris, 308:1(1989), 79–81.zbMATHMathSciNetGoogle Scholar
  11. [11]
    F. Chaplais, Product Invariant Piecewise Polynomial Approximations of Signals, Available at chaplais/FTP/Preprints/PiecewisePoly.pdf.
  12. [12]
    Stoer, J. and Burlisch, R., Introduction to Numerical Analysis, Texts in Applied Mathematics, Springer-Verlag, New York, 1993.Google Scholar

Copyright information

© Editorial Board of Analysis in Theory and Applications 2006

Authors and Affiliations

  • François Chaplais
    • 1
  1. 1.Centre Automatique et SystèmesÉcole Nationale Supérieure des Mines de ParisFontainebleau CedexFrance

Personalised recommendations