Polynomial reproduction by symmetric subdivision schemes

Authors:
Nira Dyn;Kai Hormann;Malcolm A. Sabin;Zuowei Shen
Affiliations:
School of Mathematical Sciences, Tel Aviv University, Israel;Department of Informatics, Clausthal University of Technology, Germany;Computer Laboratory, University of Cambridge, United Kingdom;Department of Mathematics, National University of Singapore, Singapore
Venue:
Journal of Approximation Theory
Year:
2008

Citing 6
Cited 9

Stationary Subdivision

Stationary Subdivision
Subdivision Methods for Geometric Design: A Constructive Approach

Subdivision Methods for Geometric Design: A Constructive Approach
Polynomial generation and quasi-interlpolation in stationary non-uniform subdivision

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Construction of biorthogonal wavelets from pseudo-splines

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Stationary subdivision schemes reproducing polynomials

Computer Aided Geometric Design
A family of subdivision schemes with cubic precision

Computer Aided Geometric Design

Full length article: Polynomial reproduction for univariate subdivision schemes of any arity

Journal of Approximation Theory
Scalar multivariate subdivision schemes and box splines

Computer Aided Geometric Design
From approximating to interpolatory non-stationary subdivision schemes with the same generation properties

Advances in Computational Mathematics
Analysis of subdivision schemes for nets of functions by proximity and controllability

Journal of Computational and Applied Mathematics
Algebraic conditions on non-stationary subdivision symbols for exponential polynomial reproduction

Journal of Computational and Applied Mathematics
Polynomial reproduction of multivariate scalar subdivision schemes

Journal of Computational and Applied Mathematics
Generalized Lane-Riesenfeld algorithms

Computer Aided Geometric Design
Exponential polynomial reproducing property of non-stationary symmetric subdivision schemes and normalized exponential B-splines

Advances in Computational Mathematics
A unified interpolatory subdivision scheme for quadrilateral meshes

ACM Transactions on Graphics (TOG)

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Abstract

We first present necessary and sufficient conditions for a linear, binary, uniform, and stationary subdivision scheme to have polynomial reproduction of degree d and thus approximation order d+1. Our conditions are partly algebraic and easy to check by considering the symbol of a subdivision scheme, but also relate to the parameterization of the scheme. After discussing some special properties that hold for symmetric schemes, we then use our conditions to derive the maximum degree of polynomial reproduction for two families of symmetric schemes, the family of pseudo-splines and a new family of dual pseudo-splines.