Measure theoretic trigonometric functions

Abstract

We study the eigenvalues and eigenfunctions of the Laplacian ${\Delta }_{\mu }=\frac{d}{d\mu }\frac{d}{dx}$ for a Borel probability measure $\mu$ on the interval $[0,1]$ by a technique that follows the treatment of the classical eigenvalue equation $f^{\prime \prime }=-\lambda f$ with homogeneous Neumann or Dirichlet boundary conditions. For this purpose we introduce generalized trigonometric functions that depend on the measure $\mu$. In particular, we consider the special case where $\mu$ is a self-similar measure like e.g. the Cantor measure. We develop certain trigonometric identities that generalize the addition theorems for the sine and cosine functions. In certain cases we get information about the growth of the suprema of normalized eigenfunctions. For several special examples of $\mu$ we compute eigenvalues of ${\Delta }_{\mu }$ and $L_{\infty }$- and $L_2$-norms of eigenfunctions numerically by applying the formulas we developed.