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
Article

Abstract

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 

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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

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