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

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.