Generalized Taylor operators and polynomial chains for Hermite subdivision schemes

Abstract

Hermite subdivision schemes act on vector valued data that is not only considered as functions values in R r , but as consecutive derivatives, which leads to a mild form of level dependence of the scheme. Previously, we have proved that a property called spectral condition or sum rule implies a factorization in terms of a generalized difference operator that gives rise to a "difference scheme" whose contractivity governs the convergence of the scheme. But many convergent Hermite schemes, for example, those based on cardinal splines, do not satisfy the spectral condition. In this paper, we generalize the property in a way that preserves all the above advantages: the associated factorizations and convergence theory. Based on these results, we can include the case of cardinal splines and also enables us to construct new types of convergent Hermite subdivision schemes.The iteration of subdivision operators S A n , n ∈ N, is called a subdivision scheme and consists of the successive applications of level-dependent subdivision operators, acting on vector valued data, S A n : ℓ r (Z) → ℓ r (Z), defined asAn important algebraic tool for stationary subdivision operators is the symbol of the mask, which is the matrix valued Laurent polynomialWe will focus our interest on two kinds of such schemes, the first one being "traditional" vector subdivision schemes in the sense of [1], where A n is independent of n, i.e., A n (α) = A(α) for any α ∈ Z and any n ≥ 0. In the following, such schemes for which an elaborate theory of convergence exists, will simply be called a vector scheme. Their convergence is defined in the following way.Definition 1 Let S A : ℓ r (Z) → ℓ r (Z) be a vector subdivision operator. The operator is C pconvergent, p ≥ 0, if for any data g 0 ∈ ℓ r (Z) and corresponding sequence of refinements g n = S n A g 0 there exists a function ψ g ∈ C p (R, R r ) such that for any compact K ⊂ R there exists a sequence ε n with limit 0 that satisfies max α∈Z∩2 n K

A note on spectral properties of Hermite subdivision operators

“…With the factorization framework [28], regularity up to C 2 can be proved, even though from [18] the scheme S a 1 is C 3 and the scheme S a 2 is C 5 . These examples show that the spectral condition is not necessary for convergence, a fact which has also been noted recently in [24].…”

“…In this paper we study spectral properties of Hermite subdivision operators. Even though the spectral condition of an Hermite subdivision operator is not necessary for the convergence of the associated scheme [24], it plays an important role in the factorizability of the operator and the regularity of the limit [23,28]. We prove that the reproduction of polynomials with respect to a parameter τ is equivalent to the spectral condition with shifted monomials of the form (x + τ ) k /k!, extending and generalizing a result of [5].…”

“…The spectral condition is an important property for the factorizability of an Hermite subdivision scheme [23,28]. Nevertheless, it is known that it is not necessary for the convergence of a scheme [24].…”

“…This condition requires certain polynomial eigenvectors of the operator. While the spectral condition is very useful for factorizing the subdivision operator (and thus for obtaining convergence results) [23], it has recently been proved that it is not necessary for convergence of the associated scheme [24]. The paper [24] also shows how to relax the spectral condition, but still retain factorization and convergence results.…”

“…While the spectral condition is very useful for factorizing the subdivision operator (and thus for obtaining convergence results) [23], it has recently been proved that it is not necessary for convergence of the associated scheme [24]. The paper [24] also shows how to relax the spectral condition, but still retain factorization and convergence results. Furthermore, we recently showed that in general the spectral condition of order d does not imply the reproduction of polynomials up to degree d of the associated scheme [28].…”