3 M ay 2 00 4 Near - best univariate spline discrete quasi - interpolants on non - uniform partitions

Univariate spline discrete quasi-interpolants (abbr. dQIs) are approximation operators using B-spline expansions with coefficients which are linear combinations of discrete values of the function to be approximated. When working with nonuniform partitions, the main challenge is to find dQIs which have both good approximation orders and bounded uniform norms independent of the given partition. Near-best dQIs are obtained by minimizing an upper bound of the infinite norm of dQIs depending on a certain number of free parameters, thus reducing this norm. This paper is devoted to the study of some families of near-best dQIs of approximation order 2. A spline quasi-interpolant (abbr. QI) of f has the general form Qf = α∈A µ α (f)B α where {B α , α ∈ A} is a family of B-splines forming a partition of unity and {µ α (f), α ∈ A} is a family of linear functionals which are local in the sense that they only use values of f in some neighbourhood of Σ α = supp(B α). The main interest of QIs is that they provide good approximants of functions without solving any linear system of equations. In the literature, one can find the three following types of QIs: (i) Differential QIs (abbr. DQIs) : the linear functionals are linear combinations of values of derivatives of f at some point in Σ α (see e.g. [5-7]). (ii) Discrete QIs (abbr. dQIs) : the linear functionals are linear combinations of values of f at some points in some neighbourhood of Σ α (see e.g. (iii) Integral QIs (abbr. iQIs) : the linear functionals are linear combinations of weighted mean values of f in some neighbourhood of Σ α (see e.g.