标题： 关于带指数和埃尔米特权的加权拉格朗日插值的一些例子和反例 显示英文标题

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

摘要： 著名的伯恩斯坦关于经典多项式拉格朗日插值最优节点系统的猜想，在半个世纪以上的时间内未被解决，于1978年由T. Kilgore解决。 紧接着他，埃德施的附加猜想也被deBoor和Pinkus解决。 这些突破性的成果建立在一个关于某些区间极大值导数（雅可比）矩阵非奇异性的基本辅助结果之上。 在上述突破之后，人们做出了大量努力，将这些结果扩展到至少某些受限函数类和切比雪夫-哈尔子空间的情况。 在这里，我们分析在半直线上的指数加权插值或全实直线上带有赫米特权的情况下，关键辅助陈述在多大程度上仍然成立。 结果表明，进行这种非奇异性的陈述是不可能的：反例表明相应的导数矩阵也可能是奇异的。 剩下的问题是进一步研究伯恩斯坦和埃德施的特征是否仍然有效。 带有拉盖尔权的加权多项式与常数函数结合的切比雪夫-哈尔系统及其相应的插值之前已经进行了研究。 还提到了证明伯恩斯坦和埃德施猜想的一些提示。 我们的目标是详细展示新情况下所需的所有辅助结果的证明。 显示英文摘要

摘要： 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 Erd\H{o}s 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 functions and Chebyshev-Haar subspaces. Here, we analyse, in what extent the key auxiliary statement remains true in case of exponentially weighted interpolation on the halfline, or with Hermite weights on the full real line. It turns out that carrying through this nonsingularity statement is impossible: counterexamples demonstrate that the respective derivative matrices may as well be singular. It remains to further study if the fundamental Bernstein- and Erd\H{o}s characterizations remain valid. The Chebyshev--Haar system of weighted polynomials with Laguerre weigth adjoined with constant functions and the corresponding interpolation were previously studied, as well. Some hints were also mentioned for he proof of Bernstein and Erd\H{o}s conjectures. Our aim is to present in detail the proof of all the auxiliary results needed in the new setting.