arXiv:math/0602349v1 [math.NA] 16 Feb 2006 Error analysis for quadratic spline quasi-interpolants on non-uniform criss-cross triangulations of bounded rectangular domains Version 25/1/06 Catterina Dagnino Dipartimento di Matematica dell’Universit` a di Torino Via Carlo Alberto 10, 10123 Torino, Italy email: catterina.dagnino@unito.it Paul Sablonni` ere INSA de Rennes, 20 avenue des Buttes de Co¨ esmes, CS 14315, 35043 Rennes Cedex, France. email: Paul.Sablonniere@insa-rennes.fr Abstract. Given a non-uniform criss-cross partition of a rectangular domain Ω, we analyse the error between a function f defined on Ω and two types of C 1 -quadratic spline quasi- interpolants (QIs) obtained as linear combinations of B-splines with discrete functionals as coefficients. The main novelties are the facts that supports of B-splines are contained in Ω and that data sites also lie inside or on the boundary of Ω. Moreover, the infinity norms of these QIs are small and do not depend on the triangulation: as the two QIs are exact on quadratic polynomials, they give the optimal approximation order for smooth functions. Our analysis is done for f and its partial derivatives of the first and second orders and a particular effort has been made in order to give the best possible error bounds in terms of the smoothness of f and of the mesh ratios of the triangulation. MSC. 65D07; 65D10; 41A25 Keywords. Bivariate splines; Approximation by splines. 1 Introduction Given a non-uniform criss-cross partition of a rectangular domain Ω, we analyse the error between a function f defined on Ω and two C 1 quadratic spline quasi-interpolants (abbr. QIs), denoted S 2 and W ∗ 2 , obtained as linear combinations of B-splines with discrete co- efficient functionals. The first operator S 2 was described by the second author in [11][14] and the second one W ∗ 2 is a slight modification of the operator W 2 introduced by Chui and Wang in [4], and also studied by Chui and He in [2], Wang and Lu in [16] and by the first 1