A Complete Classification of Fourier Summation Formulas on the real line

TL;DR

The paper addresses the problem of classifying Fourier Summation pairs (FS-pairs) on the real line under a mild decay condition, linking the Fourier-sum identity $\int_{\mathbb{R}} \widehat{\varphi}(t)\, d\mu(t)=\sum_{n\ge0} a(\lambda_n)\varphi(\lambda_n)$ to holomorphic almost periodic functions in the generalized Nevanlinna class $\mathfrak{N}_{\le k}-\mathfrak{N}_{\le k}$. The authors construct a bijection between FS-pairs with finite degree and finite exponential growth of $a$ and such $F$, via a Bridge Lemma that connects FS-sums with a Nevanlinna integral, and provide explicit representations: $F(z)=\frac{(z^2+1)^k}{2\pi i}\int \frac{tz+1}{t-z}\frac{d\mu(t)}{(1+t^2)^{k+1}}+iQ(z)$ and $F(z)=\frac{1}{2}a(0)+\sum_{\lambda>0} a(\lambda) e^{2\pi i\lambda z}$. The paper also proves the converse under weaker conditions and furnishes concrete FS-pairs, including the Selberg Trace Formula, a crystalline measure from $r_3(n)$, and a modular-form–based Guinand family, demonstrating the framework’s breadth and potential for new identities in number theory and crystallography.

Abstract

We completely classify Fourier summation formulas of the form $$ \int_{\mathbb{R}} \widehat{\varphi}(t) dμ(t)=\sum_{n=0}^{\infty} a(λ_n)\varphi(λ_n), $$ that hold for any test function $\varphi$, where $\widehat\varphi$ is the Fourier transform of $\varphi$, $μ$ 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.

Theorems & Definitions (10)

• Definition 1: Fourier summation pair
• Theorem 1: Classification of FS-pairs
• Theorem 2
• Proposition 3
• Lemma 4: Properties of auxiliary functions
• proof
• Lemma 5: The Bridge Lemma
• proof
• proof : Proof of Theorem \ref{['thm:main']}: Necessity
• proof : Proof of Theorem \ref{['thm:converse_main']}