Regularity of non-stationary subdivision: a matrix approach.

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.