Abstract
In this paper, we study scalar multivariate non-stationary subdivision schemes with integer dilation matrix M and present a unifying, general approach for checking their convergence and for determining their Hölder regularity (latter in the case M = mI, m ge 2). The combination of the concepts of asymptotic similarity and approximate sum rules allows us to link stationary and non-stationary settings and to employ recent advances in methods for exact computation of the joint spectral radius. As an application, we prove a recent conjecture by Dyn et al. on the Hölder regularity of the generalized Daubechies wavelets. We illustrate our results with several examples.
Highlights
We provide a general, unifying method for convergence and regularity analysis of multivariate non-stationary, i.e. level-dependent, subdivision schemes with an integer dilation matrix M whose eigenvalues are all larger than 1 in the absolute value
We show that the joint spectral radius techniques are applicable for all non-stationary schemes that satisfy two mild assumptions: all level-dependent masks have the same bounded support and satisfy the so-called approximate sum rules
We show that the approximate sum rules are “almost necessary” for convergence and regularity of non-stationary schemes
Summary
We provide a general, unifying method for convergence and regularity analysis of multivariate non-stationary, i.e. level-dependent, subdivision schemes with an integer dilation matrix M whose eigenvalues are all larger than 1 in the absolute value. The Hölder regularity of subdivision limits is derived from the joint spectral radius of a finite set of linear operators which are restrictions of transition matrices of the subdivision scheme to their special linear subspace. As in the stationary case, we show how to express the Hölder regularity of non-stationary subdivision in terms of the joint spectral radius of the limit points of the sequence of transition matrices restricted to this limiting linear subspace. Both results provide a powerful tool for analysis of non-stationary subdivision schemes. For analysis and applications of Rvachev-type schemes we refer the reader, for example, to [15,33,49]
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