On some examples and counterexamples about weighted Lagrange interpolation with Exponential and Hermite weights

TL;DR

This paper investigates the nonsingularity of derivative matrices in weighted Lagrange interpolation with exponential and Hermite weights, providing counterexamples and detailed proofs to understand the validity of classical conjectures in these contexts.

Contribution

It extends the analysis of nonsingularity in polynomial interpolation to weighted cases, offering counterexamples and a comprehensive proof framework for these settings.

Findings

Counterexamples show potential singularity of derivative matrices in weighted interpolation.

The classical Bernstein and Erd"H{o}s conjectures may not hold in weighted contexts.

Detailed proofs and auxiliary results are provided for the weighted interpolation scenarios.

Abstract

The famous Bernstein conjecture about optimal node systems in classical polynomial Lagrange interpolation, standing unresolved for about half a century, was solved by T. Kilgore in 1978. Immediately following him, also the additional conjecture of Erd\H{o}s was solved by de Boor and Pinkus. These breakthrough achievements were built on a fundamental auxiliary result on nonsingularity of derivative (Jacobian) matrices of certain interval maxima in function of the nodes. After the above breakthrough, a considerable effort was made to extend the results to the case of at least certain Chebyshev-Haar spaces of functions. Here, we analyse, in what extent the key nonsingularity statement remains true in case of exponentially weighted interpolation on the halfline, or with Hermite weights on the full real line. In these settings counterexamples demonstrate that the respective derivative matrices may as well be singular. It remains to further study if the Bernstein- and Erd\H{o}s characterizations remain valid. The ``hybrid'' Chebyshev-Haar system of exponentially weighted polynomials adjoined with constant functions and the corresponding interpolation were previously studied, as well. Some hints were also given for the proof of the respective Bernstein and Erd\H{o}s conjectures. We present in detail the full proof together with all the auxiliary results needed in this setting.