Leírás
The famous Bernstein conjecture about optimal node systems for classical polynomial Lagrange interpolation, which had stood unresolved for over half a century, was solved by T. Kilgore in 1978. Immediately following him, the additional conjecture of Erdős was also solved by de Boor and Pinkus. These breakthrough achievements were built on a fundamental auxiliary result on the 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 the extent to which the key nonsingularity statement remains true in the case of exponentially weighted interpolation on the half-line 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 whether the Bernstein- and Erdős characterizations remain valid.
The Chebyshev-Haar system of exponentially weighted polynomials, adjoined
with constant functions and the corresponding interpolation, was 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 consideration.