Robert S. Strichartz
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 $x$.Analysis of the Laplacian and Spectral Operators on the Vicsek Set.
Tue, 09/08/2009 - 12:16 — SomNambulAuthors: Robert S. Strichartz, Sarah Constantin, Miles Wheeler
Subjects: Analysis of PDEs
AbstractWe study the spectral decomposition of the Laplacian on a family of fractals
$\mathcal{VS}_n$ that includes the Vicsek set for $n=2$, extending earlier
research on the Sierpinski Gasket. We implement an algorithm [24] for spectral
decimation of eigenfunctions of the Laplacian, and explicitly compute these
eigenfunctions and some of their properties. We give an algorithm for computing
inner products of eigenfunctions. We explicitly compute solutions to the heat
equation and wave equation for Neumann boundary conditions. We study gaps in
the ratios of eigenvalues and eigenvalue clusters.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.Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals.
Fri, 09/04/2009 - 12:10 — SomNambulAuthors: Robert S. Strichartz, Luke G Rogers, Alexander Teplyaev
Subjects: Classical Analysis and ODEs
AbstractWe provide two methods for constructing smooth bump functions and for
smoothly cutting off smooth functions on fractals, one using a probabilistic
approach and sub-Gaussian estimates for the heat operator, and the other using
the analytic theory for p.c.f. fractals and a fixed point argument. The heat
semigroup (probabilistic) method is applicable to a more general class of
metric measure spaces with Laplacian, including certain infinitely ramified
fractals, however the cut off technique involves some loss in smoothness. From
the analytic approach we establish a Borel theorem for p.c.f.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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