Construction Techniques for Highly Accurate Quasi-Interpolation Operators

AbstractUnder mild additional assumptions this paper constructs quasi-interpolants in the form fh(x)=∑j=−∞+∞f(hj)ϕh(xh−j),x∈R,h>0, ((0.1))with approximation order ℓ−1, whereϕh(x) is a linear combination of translatesψ(x−jh) of a functionψinCℓ(R). Thus the order of convergence of such operators can be pushed up to a limit that only depends on the smoothness of the functionψ. This approach can be generalized to the multivariate setting by using discrete convolutions with tensor products of odd-degreeB-splines