Description
The famous Bernstein conjecture about optimal node systems of classical polynomial Lagrange interpolation, standing unresolved for over half a century, was solved by T. Kilgore in 1978. Immediately following him, also the additional conjecture of Erdős was solved by deBoor 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 restricted classes of Chebyshev-Haar spaces of functions.
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 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 Bernstein and Erdős conjectures. We present the main steps of the proof, emphasizing those that demanded more careful considerations.