Laplacians on fractals with spectral gaps have nicer Fourier series

TLDR: In this article, the Sierpinski gasket and related fractals have been studied intensively using both probabilistic and analytic tools, as a rough counterpart to Laplacians on smooth Riemannian manifolds.

Abstract: On the Sierpinski gasket and related fractals, partial sums of Fourier series (spectral expansions of the Laplacian) converge along certain special subsequences. This is related to the existence of gaps in the spectrum. Laplacians on fractals have been studied intensively using both probabilistic and analytic tools, as a “rough” counterpart to Laplacians on smooth Riemannian manifolds ([B], [Ki], [S1]). This research has succeeded in establishing many “expected” analogs of results from the smooth theory, but has also turned up some startling differences. For example: there exist localized eigenfunctions [FS]; the square of a nonconstant function in the domain of the Laplacian is never in the domain of the Laplacian [BST]; the energy measure is singular [Ku]; the wave equation has infinite propagation speed [DSV]; the Weyl ratio does not have a limit [FS], [KL]; the Laplacian does not behave like a second order operator [S2]; to mention just a few. One might be tempted to say that the fractal world resembles the smooth world to some degree, but everything is worse. On the other hand, recent numerical experiments hint that when it comes to convergence of Fourier series, things might be better on fractals. To be specific, consider the standard Laplacican ∆ on the Sierpinski gasket SG. With either Dirichlet or Neumann boundary conditions, there is a complete orthonormal basis of eigenfunctions, say −∆uj = λjuj , j = 1, 2, 3, . . . , and every L function f has a Fourier series