Measure Theoretic Trigonometric Functions

TL;DR

This paper introduces measure-dependent trigonometric functions to analyze the eigenvalues and eigenfunctions of a measure-based Laplacian on [0,1], including self-similar measures like the Cantor measure.

Contribution

It develops generalized trigonometric functions and identities for measure-based Laplacians, extending classical eigenvalue analysis to fractal measures.

Findings

Eigenvalues computed for various measures

Growth behavior of eigenfunctions analyzed

Numerical methods applied to specific examples

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 $f'' = -\lambda f$ 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.