Characterisations by Refinement Masks

Refinable functions underlie the theory and constructions of wavelet systems on the one hand and the theory and convergence analysis of uniform subdivision algorithms. The regularity of such functions dictates in the context of wavelets, the smoothness of the derived wavelet system, and, in the subdivision context, the smoothness of the limiting surface of the iterative process. Since the refinable function is, in many circumstances, not known analytically, the analysis of its regularity, the convergence of subdivision algorithms, the stability and orthonormality of the shifts of refinable function must be based on the explicitly known mask. We established in this project a formula that computes, for isotropic dilation and in any number of variables, the sharp regularity of the refinable function in [1, 2], the convergence of subdivision algorithms [3, 4], the stability and orthonormality of the shifts of refinable function [3, 5], in terms of the spectral radius of the restriction the associated transfer operator to a specific invariant subspace.

  1. Amos Ron, Zuowei Shen, Kim Chuan Toh, Computing the Sobolev regularity of refinable functions by Arnoldi method, SIAM Journal on Matrix Analysis and Applications, 23 (2001) 57-76. PDF
  2. Amos Ron, Zuowei Shen, The Sobolev regularity of refinable functions, Journal of Approximation Theory, 106 (2000) 185-225. PDF
  3. Zuowei Shen, Refinable function vectors, SIAM Journal on Mathematical Analysis, 29 (1998), 235-250. PDF
  4. W. Lawton, S.L. Lee, Zuowei Shen, Convergence of multidimensional cascade algorithm, Numerische Mathematik, 78 (1998) 3, 427-438. PDF
  5. W. Lawton, S.L. Lee, Zuowei Shen, Stability and orthonormality of multivariate refinable functions, SIAM Journal on Mathematical Analysis,28, (1997), 999-1014. PDF