Factorization of Hermite subdivision operators preserving exponentials and polynomials

Conti, Costanza; Cotronei, Mariantonia; Sauer, Tomas

doi:10.1007/s10444-016-9453-4

Cited by 32 publications

(48 citation statements)

References 18 publications

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“…Remark 23 Proposition 22 shows that, in the terminology of [3], the generalized Taylor operator is a minimal annihilator for the chain V since it annihilates the chain and factors any subdivision operator that does so, too.…”

Section: Proofmentioning

confidence: 99%

Generalized Taylor operators and polynomial chains for Hermite subdivision schemes

Merrien

Sauer

2018

Numer. Math.

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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

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Section: Proofmentioning

confidence: 99%

Generalized Taylor operators and polynomial chains for Hermite subdivision schemes

Merrien

Sauer

2018

Numer. Math.

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“…It is shown in [18] that both schemes satisfy the special sum rule of order 7. Furthermore, it is easy to see that S a 1 satisfies the spectral condition of order 2 with spectral polynomials 1, x, 1 2! x 2 − 1 12 , but it does not satisfy the spectral condition of order 3.…”

Section: Spectral Condition and The Special Sum Rulementioning

confidence: 99%

“…x 2 − 1 12 , but it does not satisfy the spectral condition of order 3. Similarly, S a 2 satisfies the spectral condition of order 2 with spectral polynomials 1, x, 1 2! x 2 − 1 21 , but it does not satisfy the spectral condition of order 3.…”

Section: Spectral Condition and The Special Sum Rulementioning

confidence: 99%

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A note on spectral properties of Hermite subdivision operators

Moosmüller

2019

Computer Aided Geometric Design

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In this paper we study the connection between the spectral condition of an Hermite subdivision operator and polynomial reproduction properties of the associated subdivision scheme. While it is known that in general the spectral condition does not imply the reproduction of polynomials, we here prove that a special spectral condition (defined by shifted monomials) is actually equivalent to the reproduction of polynomials. We further put into evidence that the sum rule of order ℓ > d associated with an Hermite subdivision operator of order d does not imply that the spectral condition of order ℓ is satisfied, while it is known that these two concepts are equivalent in the case ℓ = d.

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“…Recently, some research has focused both on standard [4,16] and Hermite [1,2,5,25] schemes preserving not only polynomial, but also exponential data, that is, sequences of the form e λk : k ∈ Z . This generalization allows the generation of curves which also exhibit transcendental features.…”

Section: Introductionmentioning

confidence: 99%

Level-Dependent Interpolatory Hermite Subdivision Schemes and Wavelets

et al. 2018

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We study many properties of level-dependent Hermite subdivision, focusing on schemes preserving polynomial and exponential data. We specifically consider interpolatory schemes, which give rise to level-dependent multiresolution analyses through a prediction-correction approach. A result on the decay of the associated multiwavelet coefficients, corresponding to an uniformly continuous and differentiable function, is derived. It makes use of the approximation of any such function with a generalized Taylor formula expressed in terms of polynomials and exponentials.

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Factorization of Hermite subdivision operators preserving exponentials and polynomials

Cited by 32 publications

References 18 publications

Generalized Taylor operators and polynomial chains for Hermite subdivision schemes

Generalized Taylor operators and polynomial chains for Hermite subdivision schemes

A note on spectral properties of Hermite subdivision operators

Level-Dependent Interpolatory Hermite Subdivision Schemes and Wavelets

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