A note on spectral properties of Hermite subdivision operators

TL;DR

This paper investigates the relationship between spectral conditions and polynomial reproduction in Hermite subdivision schemes, establishing a specific spectral condition equivalent to polynomial reproduction and clarifying the limits of sum rules and spectral conditions.

Contribution

It proves that a special spectral condition involving shifted monomials is equivalent to polynomial reproduction in Hermite subdivision schemes.

Findings

A special spectral condition is equivalent to polynomial reproduction.

Sum rule of order greater than the scheme's order does not imply the spectral condition.

In the case where order equals the scheme's order, sum rule and spectral condition are equivalent.

Abstract

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 $\ell=d$.