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

摘要采样序列和插值序列在许多经典的Hilbert空间中得到了刻画，如哈代空间、Bergman空间、Fock空间、Dirichlet空间和Paley-Wiener空间，但在所有这些情况下，所考虑的函数的解析性域都是圆盘或平面。PI已经得到了在Hilbert空间上采样和插值函数的充分条件，这些函数在一大类Riemann曲面上是解析的，但在该论文中还有许多问题没有解决。有些直接涉及到采样和插值的问题，但其他人处理更一般的某些想法介绍的文章，以及它们是如何与经典不变量的开放黎曼曲面。关于多复变量域上的采样和插值序列知之甚少，但PI与其他人合作，已经获得了超曲面被采样或插值的充分条件。虽然这确实导致了一些新的条件序列，作者希望，这些想法可以应用到获得一个完整的characterization.Sampling和插值是根本的重要性，在数学和科学的大。 上面提到的函数的特殊空间在应用中很重要，原因有很多，其中最重要的是它们由具有“有限能量”的函数组成。自复分析的早期以来，函数插值问题一直是一个中心主题，而采样问题则是最近才出现的。 采样在20世纪40年代由香农推广，作为他的信息理论的重要组成部分。从那时起，工程师们就对采样问题产生了浓厚的兴趣，以至于采样的概念现在已经成为日常社会的集体意识。简而言之，对函数进行采样包括在某个足够大但离散的点集上测量函数的值。 测量应足够频繁，以便功能完全由其检测值确定。插值问题可以被看作是从这个离散的，或“数字”数据重建函数的对偶问题。这个建议的一个重要方面涉及到处理复杂分析中的开放性问题的本科生项目。一个这样的问题是问题的多项式（在几个真实的变量）的因素调和多项式。另一个问题涉及Bergman空间极大值原理中最佳常数的确定。这两个问题都可以通过学生的实验工作来收集证据，以找到适当的解决方案。

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.