Mathematical Sciences: Degree of Convergence of Eigenfunction Expansions

在这个项目中，主要的研究人员将开始初步工作，以确定扩展一维技术以研究二维傅立叶级数和更一般的Sturm-Liouville类型的本征函数展开的收敛速度的可行性。特别是，她将考虑有界变差函数和导数有界变差函数的部分和的收敛速度和渐近行为。涉及本征函数展开的问题出现在许多应用数学和工程领域。有两个基本问题需要回答。第一个涉及展开式的收敛速度，第二个涉及当项数增加到无穷大时部分和的渐近行为。首席研究员计划通过扩展在一维中适用于级数展开的技术来回答这些关于二维级数展开的问题。

In this project the principal investigator will initiate preliminary work to determine the feasibility of extending one- dimensional techniques to investigate the rates of convergence of two-dimensional Fourier series and more general eigenfunction expansions of Sturm-Liouville type. In particular, she will consider rates of convergence and the asymptotic behavior of partial sums of functions of bounded variation and functions whose derivatives have bounded variation. Problems involving eigenfunction expansions arise in a host of areas of applied mathematics and engineering. There are two basic questions that need to be answered. The first involves the rate of convergence of the expansion and the second involves the asymptotic behavior of the partial sums as the number of terms increases to infinity. The principal investigator plans on answering these questions for series expansions in two dimensions by extending techniques that have worked for series in one dimension.