Szegö Limit Theorems on the Sierpiński Gasket

  title={Szeg{\"o} Limit Theorems on the Sierpiński Gasket},
  author={Kasso A. Okoudjou and Luke G. Rogers and Robert S. Strichartz},
  journal={Journal of Fourier Analysis and Applications},
We use the existence of localized eigenfunctions of the Laplacian on the Sierpiński gasket (SG) to formulate and prove analogues of the strong Szegö limit theorem in this fractal setting. Furthermore, we recast some of our results in terms of equally distributed sequences. 

Some spectral properties of pseudo-differential operators on the Sierpinski Gasket

We prove versions of the strong Sz\"ego limit theorem for certain classes of pseudodifferential operators defined on the Sierpi\'nski gasket. Our results used in a fundamental way the existence of

Spectral Analysis Beyond $$\ell ^2$$ on Sierpinski Lattices

We study the spectrum of the Laplacian on the Sierpinski lattices. First, we show that the spectrum of the Laplacian, as a subset of $\mathbb{C}$, remains the same for any $\ell^p$ spaces. Second, we

Gaps in the spectrum of the Laplacian on $3N$-Gaskets

This article develops analysis on fractal $3N$-gaskets, a class of post-critically finite fractals which include the Sierpinski triangle for $N=1$, specifically properties of the Laplacian $\Delta$


. We study the spectrum of the Laplacian on the Sierpinski lattices. First, we show that the spectrum of the Laplacian, as a subset of C , remains the same for any (cid:96) p spaces. Second, we

Spectral dimension and Bohr's formula for Schrödinger operators on unbounded fractal spaces

We establish an asymptotic formula for the eigenvalue counting function of the Schrödinger operator − Δ + V ?> for some unbounded potentials V on several types of unbounded fractal spaces. We give

Spectral analysis on infinite Sierpiński fractafolds

A fractafold, a space that is locally modeled on a specified fractal, is the fractal equivalent of a manifold. For compact fractafolds based on the Sierpiński gasket, it was shown by the first author

Eigenvalues of Laplacians on Higher Dimensional Vicsek Set Graphs

We study the graphs associated with Vicsek sets in higher dimensional settings. First, we study the eigenvalues of the Laplacians on the approximating graphs of the Vicsek sets, finding a general

Nontangential Limits and Fatou-Type Theorems on Post-Critically Finite Self-Similar Sets

AbstractIn this paper we study the boundary limit properties of harmonic functions on ℝ+×K, the solutions u(t,x) to the Poisson equation $$\frac{\partial^2 u}{\partial t^2} + \Delta u = 0,$$ where K

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincaré inequality and a weak Bakry–Émery



On a spectral analysis for the Sierpinski gasket

A complete description of the eigenvalues of the Laplacian on the finite Sierpinski gasket is presented. We then demonstrate highly oscillatory behaviours of the distribution function of the

Asymptotics of eigenvalue clusters for Schrödinger operators on the Sierpinski gasket

In this note we investigate the asymptotic behavior of spectra of Schrodinger operators with continuous potential on the Sierpinski gasket SG. In particular, using the existence of localized

Spectral Analysis on Infinite Sierpiński Gaskets

We study the spectral properties of the Laplacian on infinite Sierpin ski gaskets. We prove that the Laplacian with the Neumann boundary condition has pure point spectrum. Moreover, the set of

Fractal differential equations on the Sierpinski gasket

Let Δ denote the symmetric Laplacian on the Sierpinski gasket SG defined by Kigami [11] as a renormalized limit of graph Laplacians on the sequence of pregaskets Gm whose limit is SG. We study the


In this article a problem in the theory of Toeplitz forms is analyzed. The problem was first formulated and solved by G. Szeg? in 1952, and since solved by many authors under more general conditions.

Szegö Type Limit Theorems

The operators Pk coincide with the spectral projections Pλ of the selfadjoint operator (− d/dx) in L2(S) corresponding to the intervals [0, λ) with k < λ 6 k + 1. Following V. Guillemin [G], we

Function spaces on fractals

Taylor Approximations on Sierpinski Gasket Type Fractals

For a class of fractals that includes the familiar Sierpinski gasket, there is now a theory involving Laplacians, Dirichlet forms, normal derivatives, Green's functions, and the Gauss–Green

Harmonic calculus on p.c.f. self-similar sets

The main object of this paper is the Laplace operator on a class of fractals. First, we establish the concept of the renormalization of difference operators on post critically finite (p.c.f. for

The analogue of the strong Szegö limit theorem on the 2- and 3-dimensional spheres

This theorem was first proved by G. Szego [3] for positive functions f in the class C with α > 0. Conditions on f were relaxed by several people (see, for example, [7] and [4]), until the sharp