RUI: Sampling and Interpolation on Riemann Surfaces and in Several Complex Variables

许多经典的解析函数的Hilbert空间，如Hardy、Bergman、Fock、Dirichlet和Paley-Wiener空间的采样和内插序列已被刻画，但在所有这些情况下，所考虑的函数的可解析域要么是圆盘，要么是平面。PI得到了一大类Riemann曲面上解析函数在Hilbert空间上采样和插补的充分条件，但文中仍有许多问题有待解决。其中一些直接与采样和插补问题有关，但另一些则更一般地处理文章中介绍的某些概念，以及它们与开放黎曼曲面的经典不变量的关系。关于多复变区域上的采样和插值序列，人们知之甚少，但PI与其他人合作，得到了超曲面采样或插值的充分条件。虽然这确实导致了关于序列的一些新的条件，但作者希望这些想法可以被应用于获得完整的特征。采样和内插在数学和整个科学中具有基本的重要性。由于许多原因，上述函数的特定空间在应用中是重要的，其中最重要的原因是它们由具有“有限能量”的函数组成。自复杂分析的早期以来，函数的内插问题一直是一个中心主题，而抽样问题则是最近出现的。抽样是香农在20世纪40年代作为他的信息理论的重要组成部分而普及的。从那时起，工程师们一直对抽样问题非常感兴趣，以至于抽样的概念现在已经进入了日常社会的集体意识中。简而言之，对函数进行采样包括测量函数在某一足够大但离散的点集上的值。测量应足够频繁，以便函数完全由其检测值确定。插值法的问题可以看作是从离散的，或“数字的”数据重建函数的对偶问题。一个这样的问题是，哪些多项式(在多个实数变量中)是调和多项式的因式。另一类是确定Bergman空间极大值原理中的最佳常数。这两个问题都适用于学生的实验工作，以收集证据以获得适当的猜想解决方案。

ABSTRACTSampling and interpolation sequences have been characterized for many classical Hilbert spaces of analytic functions, such as the Hardy, Bergman, Fock, Dirichlet and Paley-Wiener space, but in all of these cases the domains of analyticity of the functions under consideration are either the disk or plane. The PI has obtained sufficient conditions for sampling and interpolation on Hilbert spaces of functions that are analytic on a large class of Riemann surfaces, but many questions are left open in that paper. Some are directly related to the question of sampling and interpolation, but others deal more generally with certain ideas introduced in the article and how they are related to classical invariants of open Riemann surfaces. Very little is known about sampling and interpolation sequences on domains in several complex variables, but the PI, in collaboration with others, has obtained sufficient conditions for hypersurfaces to be sampling or interpolating. While this does lead to some new conditions on sequences, the authors are hopeful that the ideas can be applied to obtain a complete characterization.Sampling and interpolation are of fundamental importance in mathematics and in the sciences at large. The particular spaces of functions mentioned above are important in applications for many reasons, the most significant of which is that they consist of functions with ``finite energy''. The problem of the interpolation of functions has been a central theme since the early days of complex analysis, while the sampling problem is considerably more recent. Sampling was popularized by Shannon in the 1940s as a significant portion of his theory of information. Since that time, engineers have been deeply interested in matters of sampling, so much so that the notion of sampling is now in the collective consciousness of everyday society. In a nutshell, sampling a function consists of measuring the value of a function at a certain sufficiently large but discrete set of points. The measurements should be made often enough so that the function is completely determined by its detected values. The interpolation problem can then be seen as the dual problem of reconstructing the function from this discrete, or ``digital'' data.An important aspect of this proposal involves undergraduate student projects dealing with open problems in Complex Analysis. One such problem is the question of which polynomials (in several real variables) are factors of harmonic polynomials. Another involves the determination of the best constant in the Bergman space maximum principle. Both of these questions are amenable to experimental work by students to gather evidence for an appropriate conjectured solution.