Two results on polynomial interpolation in equally spaced points
The authors present two results on the behavior of polynomial interpolation in equally spaced points which, although not new, are, in the authors' opinion, generally unknown to approximation theorists. First, if \(p_ n(t)\) is the polynomial interpolant to \(e^{i\alpha x}\), \(\alpha\in\mathbb{R}\), in the points \(0,1,\dots,n\), then \(\| f-p_ n\|_{[0,n]}\to 0\) as \(n\to\infty\) if and only if \(|\alpha|\leq\pi/3\). Second, the Lebesgue constant \(\Lambda_ n\) (supremum norm of the \(n\)-the interpolation operator) satisfies \(\Lambda^{1/n}_ n\to 2\) as \(n\to\infty\). Both theorems are proved by the use of the Newton interpolation formula.
scientific article; zbMATH DE number 12878
scientific article