矩形节点集的Birkhoff插值
Birkhoff Interpolation with Rectangular Sets of Nodes
AI总结：
本文研究以矩形节点集（笛卡尔网格）为主的Birkhoff插值，通过描述多个正则性准则并举例说明，部分实现了对可解性的几何理解，并提出了若干猜想。
摘要：
尽管在理论和应用中都很重要，但以节点集形状为主要关注点的Birkhoff插值理论仍然缺失。虽然我们将考虑各种形状（例如，我们找出所有使得相关Lagrange问题有唯一解的形状），但我们将集中于最简单的形状之一：“矩形”（也称为“笛卡尔网格”）。最终目标是获得对可解性的几何理解。我们通过描述几个正则性准则部分实现了这一点，并用许多例子加以说明。最后，我们讨论了一些猜想，我们认为这些猜想对于理解高维Birkhoff插值方案的行为很重要。尽管我们在许多不相关的案例中证明了这些猜想，但我们相信“完整证明”需要超越插值理论通常方法的新思想（可能涉及代数几何或代数拓扑等领域）。
英文摘要：
Although it is important both in theory as well as in applications, a theory of Birkhoff interpolation with main emphasis on the shape of the set of nodes is still missing. Although we will consider various shapes (e.g. we find all the shapes for which the associated Lagrange problem has unique solution), we concentrate on one of the simplest shapes:``rectangular'' (also called "cartesian grids"). The ultimate goal is to obtain a geometrical understanding of the solvability. We partially achieve this by describing several regularity criteria, which we illustrate by many examples. At the end we discuss several conjectures which, we think, are important in understanding the behaviour of Birkhoff interpolation schemes in higer dimensions. Although we prove these conjectures in many unrelated cases, we believe that a ``complete proof'' requires new ideas which go beyond the usual methods in interpolation theory (and may reach areas such as algebraic geometry or algebraic topology).
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