A Complete Classification of Fourier Summation Formulas on the real line

Abstract

We completely classify Fourier summation formulas of the form $${\int }_{\mathbb{R}}\hat{\phi }(t)d\mu (t)=\sum _{n=0}^{\infty }a({\lambda }_n)\phi ({\lambda }_n),$$ that hold for any test function $\phi$, where $\hat{\phi }$ is the Fourier transform of $\phi$, $\mu$ is a fixed complex measure on $\mathbb{R}$ and $a:\{{\lambda }_n{\}}_{n\ge 0}\to \mathbb{C}$ is a fixed function. We only assume the decay condition $${\int }_{\mathbb{R}}\frac{d|\mu |(t)}{(1+t^2{)}^{c_1}}+\sum _{n\ge 0}|a({\lambda }_n)|e^{-c_2|{\lambda }_n|}<\infty ,$$ for some $c_1,c_2>0$. This completes the work initiated by the first author previously, where the condition $c_1\le 1$ was required. We prove that any such pair $(\mu ,a)$ 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 $(\mu ,a)$. We provide important examples of such summation formulas not contemplated by the previous results, such as Selberg's trace formula.

19 pages