An algebraic approach to polynomial reproduction of Hermite subdivision schemes
Provides theoretical foundations for polynomial reproduction in Hermite subdivision, relevant to geometric modeling and approximation theory.
The paper derives algebraic conditions for symbols of Hermite subdivision schemes to reproduce polynomials, ensuring the spectral condition. Examples confirm the conditions for primal and dual schemes.
We present an accurate investigation of the algebraic conditions that the symbols of a univariate, binary, Hermite subdivision scheme have to fulfil in order to reproduce polynomials. These conditions are sufficient for the scheme to satisfy the so called spectral condition. The latter requires the existence of particular polynomial eigenvalues of the stationary counterpart of the Hermite scheme. In accordance with the known Hermite schemes, we here consider the case of a Hermite scheme dealing with function values, first and second derivatives. Several examples of application of the proposed algebraic conditions are given in both the primal and the dual situation.