Applied Numerical Mathematics 57 (2007) 962–973 www.elsevier.com/locate/apnum Recursive computation of bivariate Hermite spline interpolants ✩ A. Mazroui, D. Sbibih ∗ , A. Tijini Département de Mathématiques et Informatique, Faculté des Sciences, Université Mohammed I, Oujda, Morocco Available online 24 October 2006 Abstract Let u be a function defined on a triangulated bounded domain Ω in R 2 . In this paper, we study a recursive method for the construction of a Hermite spline interpolant u k of class C k on Ω, defined by some data scheme D k (u). We show that when D r −1 (u) ⊂ D r (u) for all 1  r  k, the spline function u k can be decomposed as a sum of (k + 1) simple elements. As application, we give the decomposition of the Ženišeck polynomial spline of class C k and degree 4k + 1, and we illustrate our results by an example.  2006 IMACS. Published by Elsevier B.V. All rights reserved. MSC: 41A05; 41A15; 65D05; 65D07; 65D10 Keywords: Interpolation; Bivariate Hermite splines; Decomposition The motivation for the study of piecewise polynomial functions is provided by their potential for applications. They can and have been used for the interpolation and approximation of data and the design of curves and surfaces. The utilization of such functions needs to determine the structure of their corresponding spaces. With regard to this topic, a significant theory has been built in the two-dimensional case. More specifically, if we denote by τ a triangulation of a bounded domain Ω of R 2 and by P k n (Ω,τ) the space of piecewise polynomial functions of total degree n and of class C k on Ω , then it is well known, see [2], that for n  4k + 1 there exists a locally supported basis of the space for interpolation schemes. This result was extended in [10] for the case n  3k + 2 and later, some other interesting results related to such a spline space was developed in the literature, see e.g. [1,3,4,6–8] and the references therein. The purpose of this paper is to describe a new method for computing recursively bivariate Hermite spline interpolants. The well known methods for building these classical interpolants are based on the Hermite fundamental functions. But, the lack of recursive formulae for computing these basis functions makes this construction rather complicated. In order to overcome this difficulty, we have proposed in [13] (see also [14]) a simple method allowing us to compute recursively an univariate Hermite spline interpolant of class C k and degree 2k + 1 of a function f defined on an interval [a,b]. More precisely, if f k is such interpolant, we have proved that f k can be decomposed in the form f k = f 0 + g 1 +···+ g k , where f 0 is the piecewise linear interpolant to f , and g r ,1  r  k , are particular splines of class C r −1 and degree 2r + 1, that satisfy interesting properties. The simplicity and the multiresolution structure of ✩ Research supported by PROTARS III D11/18. * Corresponding author. E-mail addresses: mazroui@sciences.univ-oujda.ac.ma (A. Mazroui), sbibih@sciences.univ-oujda.ac.ma (D. Sbibih), tijini@sciences.univ-oujda.ac.ma (A. Tijini). 0168-9274/$30.00  2006 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2006.09.004