Orthogonal Polynomials with Respect to Self-Similar Measures (2009)
Heilman, Steven M., Owrutsky, Philip, Strichartz, Robert S.
We 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...
Analysis of the Laplacian and Spectral Operators on the Vicsek Set (2009)
Constantin, Sarah, Strichartz, Robert S., Wheeler, Miles
We 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...
Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals (2009)
Rogers, Luke G, Strichartz, Robert S., Teplyaev, Alexander
We 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...
Localized Eigenfunctions: Here You See Them, There You Don't (2009)
Heilman, Steven M., Strichartz, Robert S.
This 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 (2009)
Heilman, Steven M., Strichartz, Robert S.
Consider 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...
Outer Approximation of the Spectrum of a Fractal Laplacian (2009)
Berry, Tyrus, Heilman, Steven M., Strichartz, Robert S.
We present a new method to approximate the Neumann spectrum of a Laplacian on a fractal K in the plane as a renormalized limit of the Neumann spectra of the standard Laplacian on a sequence of...
Distribution theory on p.c.f. fractals (2009)
Rogers, Luke G., Strichartz, Robert S.
We construct a theory of distributions in the setting of analysis on post-critically finite self-similar fractals, and on fractafolds and products based on such fractals. The results include basic...
Smooth bumps, a Borel theorem and partitions of unity on p.c.f. fractals (2009)
Luke G. Rogers, Robert S. Strichartz, Er Teplyaev
Recent years have seen considerable developments in the theory of analysis on certain fractal sets from both probabilistic and analytic viewpoints [1, 10, 19]. In this theory, either a Dirichlet...
Generalized eigenfunctions and a Borel theorem on the Sierpinski gasket (2009)
Kasso A. Okoudjou, Luke G. Rogers, Robert S. Strichartz
There is a well developed theory (see [5, 9]) of analysis on certain types of fractal sets, of which the Sierpinski Gasket (SG) is the simplest non-trivial example. In this theory the fractals are...
The resolvent kernel for PCF self-similar fractals (2008)
Ionescu, Marius, Pearse, Erin P. J., Rogers, Luke G., Ruan, Huo-Jun, Strichartz, Robert S.
For the Laplacian $\Delta$ defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions, and also with Neumann...
Szego limit theorems on the Sierpinski gasket (2008)
Okoudjou, Kasso A., Rogers, Luke G., Strichartz, Robert S.
We use the existence of localized eigenfunctions of the Laplacian on the Sierpinski gasket to formulate and prove analogues of the strong Szego limit theorem in this fractal setting. Furthermore, we...
Cauchy Transforms of Self-Similar Measures (2007)
John-Peter Lund, Robert S. Strichartz, Jade P. Vinson
this paper we will be concerned with selfsimilar measures ¯. These are solutions of selfsimilar identities ¯ =
Gradients of Laplacian Eigenfunctions on the Sierpinski Gasket (2007)
DeGrado, Jessica L., Rogers, Luke G., Strichartz, Robert S.
We use spectral decimation to provide formulae for computing the harmonic gradients of Laplacian eigenfunctions on the Sierpinski Gasket. These formulae are given in terms of special functions that...
Outer boundaries of self-similar tiles (2005)
Drenning, Shawn, Palagallo, Judith, Price, Thomas, Strichartz, Robert S.
There are many examples of self-similar tiles that are connected, but whose interior is disconnected. For such tiles we show that the boundary of a component of the interior may be decomposed into a...
Calculus on the Sierpinski Gasket I: Polynomials, Exponentials and Power Series (2003)
Needleman, Jonathan, Strichartz, Robert S., Teplyaev, Alexander
We study the analog of power series expansions on the Sierpinski gasket, for analysis based on the Kigami Laplacian. The analog of polynomials are multiharmonic functions, which have previously been...
Sampling on the Sierpinski Gasket (2003)
Oberlin, Richard, Street, Brian, Strichartz, Robert S.
We study regular and irregular sampling for functions defined on the Sierpinski Gasket (SG), where we interpret "bandlimited'' to mean the function has a finite expansion in the first {\small $d_m$}...
Constructing a Laplacian on the Diamond Fractal (2001)
Kigami, Jun, Strichartz, Robert S., Walker, Katharine C.
Kigami has shown how to construct Laplacians on certain self-similar fractals, first for the Sierpiński gasket and then for the class of postcritically finite (\pcf\) fractals, subject to the...
Sampling Theory for Functions with Fractal Spectrum (2001)
Huang, Nina N., Strichartz, Robert S.
We investigate in greater detail a sampling formula given by the first author for functions whose spectrum lies in a Cantor set $K$ of a special type introduced by Jorgensen and Pedersen, where the...
Cauchy transforms of self-similar measures (1998)
Lund, John-Peter, Strichartz, Robert S., Vinson, Jade P.
The Cauchy transform of a measure in the plane, $$ F(z) = \frac{1}{2\pi i}\int_{\C} \frac{1}{z-w} \,d\mu(w)\hbox{,} $$ is a useful tool for numerical studies of the measure, since the measure of any...
Geometry Of Self-Affine Tiles I (1998)
Robert S. Strichartz, Yang Wang
. For a self--similar or self--affine tile in R n we study the following questions: 1) What is the boundary? 2) What is the convex hull? We show that the boundary is a graph directed self--affine...
Geometry Of Self--Affine Tiles Ii (1998)
Richard Kenyon Jie, Jie Li, Robert S. Strichartz, Yang Wang
. We continue the study in part I of geometric properties of self--similar and self--affine tiles. We give some experimental results from implementing the algorithm in part I for computing the...
Geometry Of Self-Affine Tiles II (1998)
Richard Kenyon, Jie Li, Robert S. Strichartz, Yang Wang
. We continue the study in part I of geometric properties of self--similar and self--affine tiles. We give some experimental results from implementing the algorithm in part I for computing the...
Densities of self-similar measures on the line (1995)
Strichartz, Robert S., Taylor, Arthur, Zhang, Tong
We describe algorithms to compute self-similar measures associated to iterated function systems (i.f.s.) on an interval, and more general self-replicating measures that include Hausdorff measure on...
Numerical experiments in Fourier asymptotics of Cantor measures and wavelets (1992)
Janardhan, Prem, Rosenblum, David, Strichartz, Robert S.
We discuss the asymptotic behavior of Fourier transforms of Cantor measures and wavelets, and related functions that might be called multiperiodic because they satisfy a simple recursion relation...