Quadratic Spline Quasi - Interpolants on Bounded Domains

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 uniform or nonuniform partition of the domainwhere the function f is to be approximated. Beyond the simplicity of their evaluation, these operators are uniformly bounded independently of the given partition and they provide the best approximation order to smooth functions. We also give some applications to various fields in numerical approximation.