Lagrange–Hermite interpolation on the real semiaxis

Abstract

In order to approximate continuous functions on [0, +∞), we consider a Lagrange-Hermite polynomial, interpolating a finite section of the function at the zeros of some orthogonal polynomials and, with its first (r − 1) derivatives, at the point 0. We give necessary and sufficient conditions on the weights for the uniform boundedness of the related operator. Moreover, we prove optimal estimates for the error of this process in the weighted L p and uniform metric.

Averaged cubature schemes on the real positive semiaxis

“…All the above formulae are constructed as tensor product of the corresponding univariate formulae and then are based on the zeros of polynomials orthogonal with respect to a weight function of the type u(x) = x α e −x . Regarding research perspectives, we want to construct these formulae by considering the more general weight u(x) = x e −x [27][28][29]. In this case, the coefficients of the recurrence relation are not known in a closed form and then must be evaluated numerically.…”