An algebraic approach to polynomial reproduction of Hermite subdivision schemes

Conti, Costanza; Hüning, Svenja

doi:10.1016/j.cam.2018.08.009

Cited by 18 publications

(14 citation statements)

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“…Recently, Han (2021) characterized univariate Hermite masks and the convergence of univariate Hermite subdivision schemes without factorizing univariate Hermite masks. Further related topics have been addressed in Conti & Hüning (2019); Jeong & Yoon (2019); Moosmüller & Dyn (2019) and references therein. For the ordered multiset Λ = {0, 0, .…”

Section: Introduction and Motivationsmentioning

confidence: 99%

Multivariate Generalized Hermite Subdivision Schemes

Han¹

2021

Preprint

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Hermite interpolation and more generally Birkhoff interpolation are useful in mathematics and applied sciences. Due to their many desired properties such as interpolation, smoothness, short support and spline connections of basis functions, multivariate Hermite subdivision schemes employ fast numerical algorithms for geometrically modeling curves/surfaces in CAGD and isogeometric analysis, and for building Hermite wavelet methods in numerical PDEs and data sciences. In contrast to recent extensive study of univariate Hermite subdivision schemes, multivariate Hermite subdivision schemes are barely analyzed yet in the literature. In this paper we first introduce a notion of generalized Hermite subdivision schemes including all Hermite and other subdivision schemes as special cases. Then we analyze and characterize generalized Hermite masks, convergence and smoothness of generalized Hermite subdivision schemes with or without interpolation properties. We also introduce the notion of linear-phase moments for generalized Hermite subdivision schemes to have the polynomial-interpolation property. We constructively prove that there exist convergent smooth generalized Hermite subdivision schemes (including Hermite, Lagrange or Birkhoff subdivision schemes) with linear-phase moments such that their basis vector functions are spline functions in C m (R d ) for any given m ∈ N and have linearly independent integer shifts. Our results not only extend and analyze many variants of Hermite subdivision schemes but also resolve and characterize as byproducts convergence and smoothness of Hermite, Lagrange or Birkhoff subdivision schemes. Even in dimension one our results significantly generalize and extend many known results on univariate Hermite subdivision schemes. Examples are provided to illustrate the theoretical results in this paper.

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Section: Introduction and Motivationsmentioning

confidence: 99%

Multivariate Generalized Hermite Subdivision Schemes

Han¹

2021

Preprint

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“…3. Theorem 9 also explains why the algebraic conditions equivalent to the reproduction of polynomials up to order 1 derived in [4] are the same as the algebraic conditions equivalent to the spectral condition of order 1 derived in [27].…”

Section: Spectral Condition and Polynomial Reproductionmentioning

confidence: 80%

“…We study parametrizations of Hermite subdivision schemes [4,5,20], which are characterized by a parameter τ ∈ R. We say that an Hermite subdivision scheme is parametrized by a parameter τ ∈ R if we consider the vector c [n] j to be attached to the value j+τ 2 n , j ∈ Z, n ∈ N. The choice τ = 0 is called the primal parametrization and τ = −1/2 is called the dual parametrization.…”

Section: Definition 1 (Subdivision Operator) a Subdivision Operator mentioning

confidence: 99%

“…4. In the case d = 1 and d = 2 and ℓ ≥ d the algebraic conditions equivalent to the reproduction of polynomials up to order ℓ of [4] now also describe the spectral condition of order ℓ with spectral polynomials (x + τ ) k /k!, k = 0, . .…”

Section: Spectral Condition and Polynomial Reproductionmentioning

confidence: 99%

“…The polynomial reproduction order of an Hermite scheme can be checked, for example, with the algebraic conditions for the mask provided in [4]. We prove in the first part of this paper that a special spectral condition (namely, the spectral condition with shifted monomials of the form (x+τ ) k k! )…”

Section: Introductionmentioning

confidence: 98%

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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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Multivariate Generalized Hermite Subdivision Schemes

Han

2023

Constr Approx

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An algebraic approach to polynomial reproduction of Hermite subdivision schemes

Cited by 18 publications

References 18 publications

Multivariate Generalized Hermite Subdivision Schemes

Multivariate Generalized Hermite Subdivision Schemes

A note on spectral properties of Hermite subdivision operators

Multivariate Generalized Hermite Subdivision Schemes

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