Regularity of Multivariate Refinable Functions

Matrix-valued subdivision schemes for generating surfaces with extraordinary vertices

“…Hence, the order of smoothness of the target surface is determined by that of the refinable function φ. If this refinable function is not a compactly supported piecewise polynomial with prescribed smoothness joining property (called a bivariate spline), the order of smoothness of φ can be analyzed by applying the theory of shift-invariant spaces [3,8,15,19,17,27].…”

Wavelet families of increasing order in arbitrary dimensions

“…This combines results relating the stability of the wavelet basis to the Sobolev regularity of the functions [12] and results relating the Sobolev regularity to the spectrum of the transfer operator [9], [25], [40]. If the filters are FIR, then this condition can be checked by computing the eigenvalues of a finite matrix, the size of which depends on the length of the filters.…”

“…To actually compute the Sobolev regularity, we need to find the transfer operator and its invariant submatrix Then we compute the eigenvalues of and use the fact that an estimate of the lower bound on the Sobolev exponent is given by [9] (8)…”

“…Definition 3: A filter is a Neville filter of order with shift if for (9) Note that τ need not be an integer vector. Consider a simple example of a linear polynomial sampled on the integers We show that the filter with is a Neville filter of order with shift…”

Matrix refinement equations: Continuity and smoothness

“…The basic question on (1.1) is how to establish the existence of continuous and smooth solutions of (1.1) with compact support in terms of its coefficients. There are three major approaches to this question: the Fourier method (the frequency domain approach) ( [2], [3]), the iteration method (the time domain approach) ( [6], [7]) and the subdivision method [1]. In this paper we use the second to obtain several criteria.…”