Spectral Analysis of Laplacians on Certain Fractals

Abstract

Surprisingly, Fourier series on certain fractals can have better convergence properties than classical Fourier series. This is a result of the existence of gaps in the spectrum of the Laplacian. In this work we prove a general criterion for the existence of gaps. Most of the known examples on which the Laplacians admit spectral decimation satisfy the criterion. Then we analyze the infinite family of Vicsek sets, finding an explicit formula for the spectral decimation functions in terms of Chebyshev polynomials. The Laplacians on this infinite family of fractals are also shown to satisfy our criterion and thus have gaps in their spectrum.