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Abstract. 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. §1.Introduction A spline quasi-interpolant (abbr. QI) of f has the general formwhere {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:

International audienceSpline quasi-interpolants are local approximating operators for functions or discrete data. We consider the construction of discrete and integral spline quasi-interpolants on uniform partitions of the real line having small infinite norms. We call them near minimally normed quasi-interpolants: they are exact on polynomial spaces and minimize a simple upper bound of their infinite norms. We give precise results for cubic and quintic quasi-interpolants. Also the quasi-interpolation error is considered, as well as the advantage that these quasi-interpolants present when approximating functions with isolated discontinuities

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