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Robert S. Strichartz

Publications and source records attributed to Robert S. Strichartz.

46 records · Page 3Linked to original sources
arXiv2009

Orthogonal Polynomials with Respect to Self-Similar Measures

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 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 $x$.

math.CA↗
arXiv2009

Analysis of the Laplacian and Spectral Operators on the Vicsek Set

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 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. We give an explicit formula for the Green's function on $\mathcal{VS}_n$. Finally, we explain how the spectrum of the Laplacian on $\mathcal{VS}_n$ converges as $n \to \infty$ to the spectrum of the Laplacian on two crossed lines (the limit of the sets $\mathcal{VS}_n$.)

math.AP↗
arXiv2009

Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals

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 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. fractals, showing that to any prescribed jet at a junction point there is a smooth function with that jet. As a consequence we prove that on p.c.f. fractals smooth functions may be cut off with no loss of smoothness, and thus can be smoothly decomposed subordinate to an open cover. The latter result provides a replacement for classical partition of unity arguments in the p.c.f. fractal setting.

math.CA↗
arXiv2009

Homotopies of Eigenfunctions and the Spectrum of the Laplacian on the Sierpinski Carpet

Consider a family of bounded domains $Ω_{t}$ in the plane (or more generally any Euclidean space) that depend analytically on the parameter $t$, and consider the ordinary Neumann Laplacian $Δ_{t}$ on each of them. Then we can organize all the eigenfunctions into continuous families $u_{t}^{(j)}$ with eigenvalues $λ_{t}^{(j)}$ also varying continuously with $t$, although the relative sizes of the eigenvalues will change with $t$ at crossings where $λ_{t}^{(j)}=λ_{t}^{(k)}$. We call these families homotopies of eigenfunctions. We study two explicit examples. The first example has $Ω_{0}$ equal to a square and $Ω_{1}$ equal to a circle; in both cases the eigenfunctions are known explicitly, so our homotopies connect these two explicit families. In the second example we approximate the Sierpinski carpet starting with a square, and we continuously delete subsquares of varying sizes. (Data available in full at www.math.cornell.edu/~smh82)

math.AP↗
arXiv2009

The resolvent kernel for PCF self-similar fractals

For the Laplacian $Δ$ 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 boundary conditions. That is, we construct a symmetric function $G^{(λ)}$ which solves $(λ\mathbb{I} - Δ)^{-1} f(x) = \int G^{(λ)}(x,y) f(y) dμ(y)$. The method is similar to Kigami's construction of the Green kernel in \cite[\S3.5]{Kig01} and is expressed as a sum of scaled and "translated" copies of a certain function $ψ^{(λ)}$ which may be considered as a fundamental solution of the resolvent equation. Examples of the explicit resolvent kernel formula are given for the unit interval, standard Sierpinski gasket, and the level-3 Sierpinski gasket $SG_3$.

math.AP↗
arXiv2009

Distribution theory on p.c.f. fractals

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 properties of test functions and distributions, a structure theorem showing that distributions are locally-finite sums of powers of the Laplacian applied to continuous functions, and an analysis of the distributions with point support. Possible future applications to the study of hypoelliptic partial differential operators are suggested.

math.FA↗
arXiv2008

Szego limit theorems on the Sierpinski gasket

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 recast some of our results in terms of equally distributed sequences.

math.SP↗
arXiv2003

Calculus on the Sierpinski Gasket I: Polynomials, Exponentials and Power Series

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 studied in connection with Taylor approximations and splines. Here the main technical result is an estimate of the size of the monomials analogous to x^n/n!. We propose a definition of entire analytic functions as functions represented by power series whose coefficients satisfy exponential growth conditions that are stronger than what is required to guarantee uniform convergence. We present a characterization of these functions in terms of exponential growth conditions on powers of the Laplacian of the function. These entire analytic functions enjoy properties, such as rearrangement and unique determination by infinite jets, that one would expect. However, not all exponential functions (eigenfunctions of the Laplacian) are entire analytic, and also many other natural candidates, such as the heat kernel, do not belong to this class. Nevertheless, we are able to use spectral decimation to study exponentials, and in particular to create exponentially decaying functions for negative eigenvalues.

math.CA↗