We describe some new univariate spline quasi-interpolants on uniform partitions of bounded intervals. Then we give some applications to numerical analysis: integration, differentiation and approximation of zeros.

We propose a new quadrature rule for Cauchy principal value integrals based on quadratic spline quasi-interpolants which have an optimal approximation order and satisfy boundary interpolation conditions. In virtue of these spline properties, we can prove uniform convergence for sequences of such quadratures and provide uniform error bounds. A computational scheme for the quadrature weights is g...

Approximation properties of the dilations of the integer translates of a smooth function, with some derivatives vanishing at infinity, are studied. The results apply to fundamental solutions of homogeneous elliptic operators and to “shifted” fundamental solutions of the iterated Laplacian. Following the approach from spline theory, the question of polynomial reproduction by quasiinterpolation i...

Quasi-interpolation is a well-known technique to construct accurate approximants to a given set of data or a given function by means of a local approach. A quasi-interpolant is usually obtained as a linear combination of a given system of blending functions that form a convex partition of unity and possess a small local support. These properties ensure both numerical stability and local control...

We study some C1 quadratic spline quasi-interpolants on bounded domains  ⊂ Rd, d = 1, 2, 3. These operators are of the form Q f (x) = ∑ k∈K () μk( f )Bk(x), where K () is the set of indices of B-splines Bk whose support is included in the domain  and μk( f ) is a discrete linear functional based on values of f in a neighbourhood of xk ∈ supp(Bk). The data points x j are vertices of a unifor...

Univariate and multivariate quadratic spline quasi-interpolants provide interesting approximation formulas for derivatives of approximated functions that can be very accurate at some points thanks to the superconvergence properties of these operators. Moreover, they also give rise to good global approximations of derivatives on the whole domain of definition. From these results, some collocatio...

A new cubature rule for a parallelepiped domain is defined by integrating a discrete blending sum of C1 quadratic spline quasi-interpolants in one and two variables. We give the weights and the nodes of this cubature rule and we study the associated error estimates for smooth functions. We compare our method with cubature rules based on tensor products of spline quadratures and classical compos...