From Hermite to stationary subdivision schemes in one and several variables

Jean-Louis Merrien (1) , Tomas Sauer (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Lehrstuhl für Numerische Mathematik

Vector and Hermite subdivision schemes both act on vector data, but since the latter one interprets the vectors as function values and consecutive derivatives they differ by the "renormalization" of the Hermite scheme in any step. In this paper we give an algebraic factorization method in one and several variables to relate any Hermite subdivision scheme that satisfies the so-called spectral condition to a vector subdivision scheme. These factorizations are natural extensions of the "zero at π" condition known for the masks of refinable functions. Moreover, we show how this factorization can be used to investigate different forms of convergence of the Hermite scheme and why the multivariate situation is conceptionally more intricate than the univariate one. Finally, we give some examples of such factorizations.

Mots clés: Subdivision; Hermite; Taylor expansion; Factorization

Domaines: Analyse numérique [math.NA]

Jean-Louis Merrien, Tomas Sauer. From Hermite to stationary subdivision schemes in one and several variables. Advances in Computational Mathematics, 2012, 36 (4), pp.547-579. ⟨10.1007/s10444-011-9190-7⟩. ⟨hal-00468245⟩