  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 $f'' = -\lambda f$ 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.
