On $p$-adic spectral measures

Abstract

A Borel probability measure $\mu$ on a locally compact group is called a spectral measure if there exists a subset of continuous group characters which forms an orthogonal basis of the Hilbert space $L^2(\mu )$. In this paper, we characterize all spectral measures in the field ${\mathbb{Q}}_p$ of $p$-adic numbers.

30 pages, 6 figures