Measures, modular forms, and summation formulas of Poisson type

TL;DR

This paper establishes a connection between Fourier eigenmeasures supported on spheres and modular-type Fourier series, leading to new Poisson-type summation formulas and extending these results to higher dimensions using Hilbert modular forms.

Contribution

It introduces the concept of $k$-spherical measures and demonstrates their correspondence with modular-type Fourier series, extending classical summation formulas to higher dimensions.

Findings

Fourier eigenmeasures on spheres relate to modular-type Fourier series.

Derived new Poisson-type summation formulas for these measures.

Extended results to higher dimensions using Hilbert modular forms.

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-Gon\c{c}alves, Lev-Reti, and Meyer, among others. Additionally, we extend our results to higher dimensions, where Hilbert modular forms yield higher-dimensional $k$-spherical measures.