A note on spectral properties of Hermite subdivision operators
This clarifies the relationship between spectral conditions and polynomial reproduction for Hermite subdivision operators, resolving a gap in the theoretical understanding for researchers in approximation theory and geometric modeling.
The paper proves that a special spectral condition (defined by shifted monomials) is equivalent to polynomial reproduction for Hermite subdivision operators, and shows that sum rules of order > d do not imply the spectral condition of that order, unlike the case ℓ = d.
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 $\ell>d$ associated with an Hermite subdivision operator of order $d$ does not imply that the spectral condition of order $\ell$ is satisfied, while it is known that these two concepts are equivalent in the case $\ell=d$.