Local Explicit Many-Knot Spline Hermite Approximation Schemes.

Abstract : Some authors considered operators of the form Omega f = sigma lambda i fN i,k, where (Ni,k) is a sequence of B-splines and (lambda i) is a sequence of linear functionals. The variation diminishing method of Schoenberg (9, 5, 6), the quasi-interpolant of de Boor and Fix are well-known. Such approximation schemes have several important advantages over spline interpolation. They can be constructed directly without matrix inversion, local error bounds can be obtained naturally. Omega i considered so-called many-knot splines which have many more knots than degrees of freedom and constructed the cardinal spline Omega f = sigma f(xi)qi,k, where qi,k is made up of B-splines on a uniform partition, has small support and satisfies qi,k(xj) = sigma ij. Such an approximation operator reproduces appropriate classes of polynomials.