In this article, we study the Fekete problem in segmental and combined nodal-segmental univariate polynomial interpolation by investigating sets of segments, or segments combined with nodes, such that the Vandermonde determinant for the respective polynomial interpolation problem is maximized. For particular families of segments, we will be able to find explicit solutions of the corresponding maximization problem. The quality of the Fekete segments depends hereby strongly on the utilized normalization of the segmental information in the Vandermonde matrix. To measure the quality of the Fekete segments in interpolation, we analyse the asymptotic behaviour of the generalized Lebesgue constant linked to the interpolation problem. For particular sets of Fekete segments we will get, similar to the nodal case, a favourable logarithmic growth of this constant.

翻译：本文研究分段及节点-分段混合单变量多项式插值中的Fekete问题，通过考察线段集合或线段与节点的组合，使得相应多项式插值问题的范德蒙行列式达到最大。针对特定线段族，我们能够找到相应最大化问题的显式解。在此过程中，Fekete线段的质量强烈依赖于范德蒙矩阵中线段信息的归一化方式。为评估Fekete线段在插值中的质量，我们分析了与该插值问题相关的广义勒贝格常数的渐近行为。类似于节点情形，特定Fekete线段集合将使得该常数呈现有利的对数增长。