A Complete Classification of Fourier Summation Formulas on the real line

Abstract

We completely classify Fourier summation formulas of the form $$ \int_{\mathbb{R}} \widehatφ(t) dμ(t)=\sum_{n=0}^{\infty} a(λ_n)φ(λ_n), $$ that hold for any test function $φ$, where $\widehatφ$ is the Fourier transform of $φ$, $μ$ is a fixed complex measure on $\mathbb{R}$ and $a:\{λ_n\}_{n\geq 0}\to\mathbb{C}$ is a fixed function. We only assume the decay condition $$ \int_{\mathbb{R}} \frac{d |μ|(t)}{(1+t^2)^{c_1}} + \sum_{n\geq 0} |a(λ_n)|e^{-c_2 |λ_n|}<\infty, $$ for some $c_1,c_2>0$. This completes the work initiated by the first author previously, where the condition $c_1\leq 1$ was required. We prove that any such pair $(μ,a)$ can be uniquely associated with a holomorphic map $F(z)$ in the upper-half space that is both almost periodic and belongs to a certain higher index Nevanlinna class. The converse is also true: For any such function $F$ it is possible to generate a Fourier summation pair $(μ,a)$. We provide important examples of such summation formulas not contemplated by the previous results, such as Selberg's trace formula.