On Berman's phenomenon for (0,1,2) Hermite-Fejér interpolation

Abstract

Given \(f\in C[-1,1]\) and \(n\) points (nodes) in \([-1,1]\), the Hermite-Fejer interpolation (HFI) polynomial is the polynomial of degree at most \(2n-1\) which agrees with \(f\) and has zero derivative at each of the nodes. In 1916, L. Fejer showed that if the nodes are chosen to be the zeros of \(T_{n}(x)\), the \(n\)th Chebyshev polynomial of the first kind, then the HFI polynomials converge uniformly to \(f\) as \(n\rightarrow\infty\). Later, D.L. Berman established the rather surprising result that this convergence property is no longer true for all \(f\) if the Chebyshev nodes are augmented by including the endpoints \(-1\) and \(1\) as additional nodes. This behaviour has become known as Berman's phenomenon. The aim of this paper is to investigate Berman's phenomenon in the setting of \((0,1,2)\) HFI, where the interpolation polynomial agrees with \(f\) and has vanishing first and second derivatives at each node. The principal result provides simple necessary and sufficient conditions, in terms of the (one-sided) derivatives of \(f\) at \(\pm 1\), for pointwise and uniform convergence of \((0,1,2)\) HFI on the augmented Chebyshev nodes if \(f\in C^{4}[-1,1]\), and confirms that Berman's phenomenon occurs for \((0,1,2)\) HFI.