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

TL;DR

This work analyzes weighted Lagrange interpolation with exponential weights on the halfline and Hermite weights on the real line, examining whether the classical Bernstein/Erdős framework extends to these settings. It demonstrates explicit counterexamples where the fundamental nonsingularity of derivative matrices fails, undermining the direct transfer of the classical five properties. The authors then develop a robust treatment for the hybrid system ${\mathcal{Y}}_n$, proving interlacing of the associated polynomials, a Markov-type inheritance for derivative roots, and nonsingularity as well as the properness of the key map $\Gamma$, thereby recovering Bernstein/Erdős-type conclusions in this setting. The results illuminate where classical interpolation theory can be extended and where new methods are required, and they provide a concrete corrigendum to previous claims while offering a viable pathway via the hybrid framework.

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 Erdős 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 Erdős 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 Erdős conjectures. We present in detail the full proof together with all the auxiliary results needed in this setting.

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Theorems & Definitions (40)

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