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Abstract. Refinement equations play an important role in computer graphics and wavelet analysis. In this paper we investigate multivariate refinement equations associated with a dilation matrix and a finitely supported refinement mask. We characterize the Lp-convergence of a subdivision scheme in terms of the p-norm joint spectral radius of a collection of matrices associated with the refinement mask. In particular, the 2-norm joint spectral radius can be easily computed by calculating the eigenvalues of a certain linear operator on a finite dimensional linear space. Examples are provided to illustrate the general theory.

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Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask. We connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators. Finally, we demonstrate that a refinable function in W k−1 1 (R s ) provides approximation order k.

Abstract. We consider solutions of a system of refinement equations written in the form φ = α∈Z a(α)φ(2 · −α), where the vector of functions φ = (φ 1 , . . . , φ r ) T is in (Lp(R)) r and a is a finitely supported sequence of r × r matrices called the refinement mask. Associated with the mask a is a linear operator Qa defined on (Lp(R)) r by Qaf := α∈Z a(α)f (2 · −α). This paper is concerned with the convergence of the subdivision scheme associated with a, i.e., the convergence of the sequence (Q n a f ) n=1,2,... in the Lp-norm. Our main result characterizes the convergence of a subdivision scheme associated with the mask a in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the L 2 -convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations.Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

In this paper we consider functional equations of the form . , φ r )T is an r × 1 vector of functions on the s-dimensional Euclidean space, a(α), α ∈ Z s , is a finitely supported sequence of r × r complex matrices, and M is an s ×s isotropic integer matrix such that lim n→∞ M −n = 0. We are interested in the question, for which sequences a will there exist a solution to the functional equation with each function φ j , j = 1, . . . , r, belonging to the Sobolev space W k p (R s )? Our approach will be to consider the convergence of the cascade algorithm. The cascade operator Q a associated with the sequence a is defined byLet 0 be a nontrivial r × 1 vector of compactly supported functions in W k p (R s ). The iteration scheme n = Q a n−1 , n = 1, 2, . . . , is called a cascade algorithm, or a subdivision scheme. Under natural assumptions on a, a feasible set of initial vectors is identified from the conditions on an initial vector implied by the convergence of the subdivision scheme. VECTOR SUBDIVISION SCHEMES IN SOBOLEV SPACES 129The formal definition of convergence in the Sobolev norm for the subdivision scheme is that the scheme will converge for any choice of initial vector from the feasible set (to the same solution ). We give a characterization for this concept of convergence in terms of the p-norm joint spectral radius of a finite collection of transition operators determined by the sequence a restricted to a certain invariant subspace. The invariant subspace is intimately connected to the Strang-Fix type conditions that determine the feasible set of initial vectors.  2002 Elsevier Science

We investigate linear independence of integer translates of a finite number of compactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some / ' space (1 g p g oo) and we are interested in bounding their /'-norms in terms of the L'-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

Starting with Hermite cubic splines as the primal multigenerator, first a dual multigenerator on R is constructed that consists of continuous functions, has small support, and is exact of order 2. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the primal and dual sides. This guarantees moment conditions of the corresponding wavelets. The concept of stable completions [CDP] is then used to construct the corresponding primal and dual multiwavelets on the interval as follows. An appropriate variation of what is known as a hierarchical basis in finite element methods is shown to be an initial completion. This is then, in a second step, projected into the desired complements spanned by compactly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algorithms are simple and efficient. Furthermore, in addition to the Jackson estimates which follow from the exactness, one can also show Bernstein inequalities for the primal and dual multiresolutions. Consequently, sequence norms for the coefficients based on such multiwavelet expansions characterize Sobolev norms • H s ([0,1]) for s ∈ (−0.824926, 2.5). In particular, the multiwavelets form Riesz bases for L 2 ([0, 1]).