An algebraic approach to polynomial reproduction of Hermite subdivision schemes

TL;DR

This paper investigates algebraic conditions for polynomial reproduction in univariate Hermite subdivision schemes, providing criteria that ensure spectral conditions and illustrating applications with examples.

Contribution

It introduces algebraic conditions that guarantee polynomial reproduction in Hermite schemes, linking them to spectral properties and demonstrating their application.

Findings

Algebraic conditions for polynomial reproduction are established.

Conditions ensure the spectral property of Hermite schemes.

Examples illustrate the application in primal and dual cases.

Abstract

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.