RUI: Interpolation and Sampling in Bergman Spaces; Factors of Harmonic Polynomials

用来表征插值序列的方法取决于上下文是Hardy空间、Bergman空间、Fock空间还是Dirichlet空间。本研究将尝试使用弱内插的概念来统一这些理论，弱内插可以在具有再现核的解析函数的希尔伯特空间的一般背景下定义。离开希尔伯特空间设置，另一个研究途径将集中在获得布洛赫空间内插序列的几何表征上。另一个研究方向涉及任意维欧几里得空间中调和多项式的多项式因子问题，适合于寻找有用证据的本科生的计算机实验工作。在原点有孤立零的多项式不能成为非零调和多项式的因数。这就提出了哪些多项式是非零调和多项式的因子的问题。在过去的两个世纪里，调和函数和解析函数一直是数学家、科学家和工程师们密切关注的主题。拉普拉斯算子的消失是谐波函数的特征，它自然地出现在科学和工程的几个领域，从热方程到电荷的分布。这个项目研究了解析函数和调和函数集合的性质，试图深入了解这些函数的行为。这个项目的一个关键部分是高级本科生的参与，他们将被介绍到数学研究的世界，希望能激励他们继续学习和从事数学方面的职业。向这些学生展示发现新数学知识的兴奋感，可能有助于培养下一代数学家。

The methods used to characterize interpolation sequences vary depending upon whether the context is the Hardy space, the Bergman space, the Fock space, or the Dirichlet space. This research will attempt to unify these theories, using the concept of weak interpolation, which can be defined in the general context of a Hilbert space of analytic functions with a reproducing kernel. Moving away from the Hilbert space setting, an additional avenue of research will focus on obtaining a geometric characterization of interpolating sequences for the Bloch space. Yet another direction of research, suitable for experimental computer work by undergraduates who could search for useful evidence, involves the question of polynomial factors of harmonic polynomials in Euclidean spaces of arbitrary dimension. A polynomial that has an isolated zero at the origin cannot be a factor of a nonzero harmonic polynomial. This raises the question of which polynomials are factors of nonzero harmonic polynomials.Harmonic and analytic functions have been the subject of intense scrutiny by mathematicians, scientists, and engineers for the past two centuries. The Laplacian operator, whose vanishing characterizes harmonic functions, appears naturally in several areas of science and engineering, ranging from the heat equation to the distribution of electric charges. This project investigates properties of collections of analytic and harmonic functions in an attempt to understand deep aspects of the behavior of these functions. A key part of this project is the involvement of advanced undergraduates, who will be introduced to the world of mathematical research in the hope of inspiring them to pursue further studies and a career in mathematics. Showing these students the excitement of discovering new mathematical knowledge may help create the next generation of mathematicians.