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Regularity of non-stationary subdivision: a matrix approach.

M Charina1, C Conti2, N Guglielmi3

  • 1University of Vienna, Vienna, Austria.

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PubMed
Summary

This study introduces a general method to analyze non-stationary subdivision schemes, determining their convergence and Hölder regularity. This approach unifies stationary and non-stationary settings, proving a conjecture on generalized Daubechies wavelets.

Keywords:
15A6039A9965D17

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Area of Science:

  • * Mathematical analysis
  • * Applied mathematics
  • * Computer-aided geometric design

Background:

  • * Subdivision schemes are fundamental in computer graphics and approximation theory.
  • * Analyzing non-stationary schemes is complex, lacking general convergence and regularity criteria.
  • * Existing methods often struggle to bridge stationary and non-stationary cases.

Purpose of the Study:

  • * To develop a unified framework for analyzing scalar multivariate non-stationary subdivision schemes.
  • * To establish general criteria for checking convergence and determining Hölder regularity.
  • * To apply these methods to prove a conjecture regarding generalized Daubechies wavelets.

Main Methods:

  • * Combining concepts of asymptotic similarity and approximate sum rules.
  • * Leveraging recent advances in computing the joint spectral radius.
  • * Developing a general approach applicable to integer dilation matrices (M).

Main Results:

  • * A unifying method for assessing convergence of non-stationary subdivision schemes is presented.
  • * A general approach for determining Hölder regularity (in the case of C^r) is established.
  • * A conjecture by Dyn et al. on the Hölder regularity of generalized Daubechies wavelets is proven.

Conclusions:

  • * The developed framework effectively links stationary and non-stationary subdivision schemes.
  • * The method provides a powerful tool for analyzing complex subdivision schemes.
  • * This work advances the understanding of regularity properties in wavelet and subdivision theory.