×

On interpolation. I. Quadrature- and mean-convergence in the Lagrange- interpolation. (English) Zbl 0016.10604

Let \(\{\xi_n\}\) be a sequence on \(n\) points from \([-1,+1]\) varying with \(n\); let \(L_n(x)\) denote the sequence of Lagrange polynomials coinciding with a given \(R\) integrable function \(f(x)\) at the points \(\xi_n\). The authors are interested in the mean convergence \[ \lim_{n \to \infty} \int_{-1}^{+1} |f(x)-L_n(x)|^p\, dx=0\tag{*} \] for \(p=2\) and \(p=1\). Let \(\xi_n\) be the zeros of the orthogonal polynomial \(p_n(x)\) of degree \(n\) corresponding to the weight function \(w(x)\geq \mu >0\). Then (*) holds with \(p=2\). The same is true if we choose for \(\xi_n\) the zeros of \(p_n(x)+A_np_{n-1}(x)+B_np_{n-2}(x)\), where \(A_n\) arbitrary real, \(B_n\leq 0\). If \(\int_{-1}^{+1} w(x) dx\) and \(\int_{-1}^{+1} w(x)^{-1}\, dx\) exist and \(\xi_n\) is defined by the zeros of the linear combination mentioned, (*) holds with \(p=1\). Finally the existence of a continuous function \(f(x)\) is proved for which (*) with \(p=2\) does not hold provided that \(\sum_{k=1}^n \int_{-1}^{+1} l_k(x)^2 \, dx\) is unbounded; here \(l_k(x)\) are the fundamental polynomials of the Lagrange interpolation corresponding to the set \(\xi_n\).
Reviewer: G.Szegö

MSC:

41A05 Interpolation in approximation theory
42A15 Trigonometric interpolation
PDFBibTeX XMLCite
Full Text: DOI