Measures, Modular Forms, and Summation Formulas of Poisson Type

Abstract

In this article, we show that Fourier eigenmeasures supported on spheres with radii given by a locally finite sequence, which we call k-spherical measures, correspond to Fourier series exhibiting a modular-type transformation behaviour with respect to the metaplectic group. A familiar subset of such Fourier series comprises holomorphic modular forms. This allows us to construct k-spherical eigenmeasures and derive Poisson-type summation formulas, thereby recovering formulas of a similar nature established by Cohn–Gonçalves, Lev–Reti, and Meyer, among others. Additionally, we extend our results to higher dimensions, where Hilbert modular forms yield higher-dimensional k-spherical measures.