Steven M. Heilman
Orthogonal Polynomials with Respect to Self-Similar Measures.
Tue, 10/06/2009 - 17:01 — SomNambulAuthors: Steven M. Heilman, Robert S. Strichartz, Philip Owrutsky
Subjects: Classical Analysis and ODEs
AbstractWe study experimentally systems of orthogonal polynomials with respect to
self-similar measures. When the support of the measure is a Cantor set, we
observe some interesting properties of the polynomials, both on the Cantor set
and in the gaps of the Cantor set. We introduce an effective method to
visualize the graph of a function on a Cantor set. We suggest a new
perspective, based on the theory of dynamical systems, for studying families
$P_{n}(x)$ of orthogonal functions as functions of $n$ for fixed values of $x$.Localized Eigenfunctions: Here You See Them, There You Don't.
Mon, 09/07/2009 - 10:33 — SomNambulAuthors: Steven M. Heilman, Robert S. Strichartz
Subjects: Analysis of PDEs
AbstractThis expository note explores Laplacian eigenfunction localization for
compact domains. We work in the context of a particular numerically determined,
localized, low frequency eigenfunction.Homotopies of Eigenfunctions and the Spectrum of the Laplacian on the Sierpinski Carpet.
Fri, 08/21/2009 - 19:51 — SomNambulAuthors: Steven M. Heilman, Robert S. Strichartz
Subjects: Analysis of PDEs
AbstractConsider a family of bounded domains $\Omega_{t}$ in the plane (or more
generally any Euclidean space) that depend analytically on the parameter $t$,
and consider the ordinary Neumann Laplacian $\Delta_{t}$ on each of them. Then
we can organize all the eigenfunctions into continuous families $u_{t}^{(j)}$
with eigenvalues $\lambda_{t}^{(j)}$ also varying continuously with $t$,
although the relative sizes of the eigenvalues will change with $t$ at
crossings where $\lambda_{t}^{(j)}=\lambda_{t}^{(k)}$. We call these families
homotopies of eigenfunctions. We study two explicit examples.
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