Local convergence of Lagrange interpolation associated with equidistant nodes
Let \(f\) be a function defined on \([-1,1]\), \(L_ n( ;\cdot)\) be the Lagrange interpolating polynomial of degree at most \(n\) in the equidistant nodes and let \(G_ s\) denote the interior of level curves \(\Gamma_ s\). The origin of convergence of this polynomial is studied. If \(f\) is bounded on \([-1,1]\) and analytic in \(G_ s\) (without requiring a stronger condition that \(f\) is analytic on \([-1,1]\)), then \(L_ n(f;\cdot)\) converges locally uniformly to \(f\) in \(G_ s\). Consequently if \(f\) is analytic at \(x=0\) and bounded on \([-1,1]\) then the Lagrange interpolants converge to \(f\) in a neighborhood of the origin. Problems to be solved in future are mentioned.
scientific article; zbMATH DE number 617504
scientific article