In this paper we shall study vector cascade algorithms and refinable function vectors with a general isotropic dilation matrix in Sobolev spaces. By investigating several properties of the initial function vectors in a vector cascade algorithm, we are able to take a relatively unified approach to study several questions such as convergence, rate of convergence and error estimate for a perturbed mask of a vector cascade algorithm in a Sobolev space W k p (R s)(1 � p � ∞, k ∈ N∪{0}). We shall characterize the convergence of a vector cascade algorithm in a Sobolev space in various ways. As a consequence, a simple characterization for refinable Hermite interpolants and a sharp error estimate for a perturbed mask of a vector cascade algorithm in a Sobolev space will be presented. The approach in this paper enables us to answer some unsolved questions in the literature on vector cascade algorithms and to comprehensively generalize and improve results on scalar cascade algorithms and scalar refinable functions to the vector case. Key words: vector cascade algorithm, vector subdivision scheme, refinable function vector, Hermite interpolant, initial function vector, error estimate, sum rules, smoothness.

Abstract. We consider the smoothness of solutions of a system of refinement equations written in the form φ = a(α)φ(2 ·−α), α∈Z where the vector of functions φ =(φ1,...,φr) T is in (Lp(R)) r and a is a finitely supported sequence of r × r matrices called the refinement mask. We use the generalized Lipschitz space Lip ∗ (ν, Lp(R)), ν>0, to measure smoothness of a given function. Our method is to relate the optimal smoothness, νp(φ), to the p-norm joint spectral radius of the block matrices Aε, ε =0,1, given by Aε =(a(ε+2α−β))α,β, when restricted to a certain finite dimensional common invariant subspace V. Denoting the p-norm joint spectral radius by ρp(A0|V,A1|V), we show that νp(φ) ≥ 1/p − log 2 ρp(A0|V,A1|V) with equality when the shifts of φ1,...,φr are stable and the invariant subspace is generated by certain vectors induced by difference operators of sufficiently high order. This allows an effective use of matrix theory. Also the computational implementation of our method is simple. When p = 2, the optimal smoothness is also given in terms of the spectral radius of the transition matrix associated with the refinement mask. To illustrate the theory, we give a detailed analysis of two examples where the optimal smoothness can be given explicitly. We also apply our methods to the smoothness analysis of multiple wavelets. These examples clearly demonstrate the applicability and practical power of our approach.

Abstract—In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a B-spline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory—including some new extensions for fractional orders—in a self-contained, accessible fashion. In particular, we prove that the B-spline component is entirely responsible for five key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multiscale differentiation property, and smoothness (regularity) of the basis functions. We also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions. This allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal. Additional results include a new peeling theory of smoothness, leading to the extended notion of wavelet differentiability in the-sense and a sharper theorem stating that smoothness implies order. Index Terms—Approximation order, Besov spaces, Hölder smoothness, multiscale differentiation, splines, vanishing moments, wavelets. I.

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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 must be based on the explicitly known mask. We establish in this paper a formula that computes, for isotropic dilation and in any number of variables, the sharp L 2 -regularity of the refinable function OE in terms of the spectral radius of the restriction of the associated transfer operator to a specific invariant subspace. For a compactly supported refinable function OE, the relevant invariant space is proved to be finite dimensional, and is completely characterized in terms of the dependence relat...

The purpose of this paper is to investigate spectral properties of the transition operator associated to a multivariate vector refinement equation and their applications to the study of smoothness of the corresponding refinable vector of functions. Let Φ = (φ1,..., φr) T be an r × 1 vector of compactly supported functions in L2(IR s) satisfying the refinement equation Φ = � α ∈ Zs a(α)Φ(M · − α), where M is an expansive integer matrix. We assume that M is isotropic, i.e., M is similar to a diagonal matrix diag(σ1,..., σs) with |σ1 | = · · · = |σs|. For µ = (µ1,..., µs) ∈ IN s 0, define. The smoothness of Φ is measured by the critical exponent σ −µ: = σ −µ1

Abstract. Necessary and sufficient conditions are given for the convergence of infinite products of matrices of complex numbers. The results are applied to the solution of periodic matrix refinement equations. Conditions are given for the solutions to be in L 2 ([0, 2π) s) and generate a multiresolution of multiplicity r. A general algorithm for constructing multidimensional periodic multiwavelets from a scaling vector which generates a multiresolution is also given. 1.

Abstract. In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W k p (Rs)(1 � p � ∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space W k p (Rs) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the Lp(1 � p � ∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector φ ∈ Lp(R s) (φ ∈ C(R s) when p = ∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν, Lp(R s)) for some ν> 0. This paper generalizes the results in [R. Q. Jia, K. S. Lau, and D. X. Zhou, J. Fourier Anal. Appl., 7 (2001), pp. 143–167] in the univariate setting to the multivariate setting. 1.

In this paper we study refinable vectors of spline functions for general knots. A characterisation is given of refinable spline spaces, i.e. spaces S of spline functions such that f ∈ S implies f( : 2 ) ∈ S. By considering bases for such spaces we characterise refinable splines whose integer translates are linearly independent and for which the number of spline functions is `maximal'. Further results are also mentioned.

. In this paper the convergence of the cascade algorithm in a Sobolev space is considered if the refinement mask is perturbed. It is proved that the cascade algorithm corresponding to a slightly perturbed mask converges. Moreover, the perturbation of the resulting limit function is estimated in terms of that of the masks. x1. Introduction In this paper we are concerned with the following problem: Given a compactly supported multivariate refinable function OE, how does perturbation of its finite refinement mask affect the convergence of the cascade algorithm? Further, if the cascade algorithm for the perturbed mask also converges, how the resulting limit function is related with OE? We say that a compactly supported function OE is M-refinable if it satisfies a refinement equation OE = X ff2ZZ s a(ff)OE(M \Delta \Gamma ff); (1:1) where the finitely supported sequence a = (a(ff)) ff2ZZ s is called the refinement mask. The s \Theta s matrix M is called a dilation matrix. We suppo...