Mathematical Sciences: Degree of Convergence of Eigenfunction Expansions

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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.