水平带形上傅里叶求和公式的分类
Classification of Fourier summation formulas on a horizontal strip
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中文总结 AI 辅助
该研究将傅里叶求和公式分类推广到复平面有限宽度带形情形，证明其与上半平面殆周期亚纯Nevanlinna函数对应，刻画了这类公式的特征并建立双向关联。
中文摘要 AI 辅助
我们将傅里叶求和公式的分类推广到测度μ支撑在复平面ℂ上有限宽度带形的情形。这一更广泛的框架涵盖了重要例子，包括塞尔伯格类（Selberg class）中函数的Guinand–Weil显式公式，这些例子超出了先前分类的范围。我们研究对所有φ∈ℂ^∞_c(ℝ)都成立的恒等式：∑_{n≥0}a(λ_n)φ(λ_n)=∫_{ℝ}φ̂(t)dν(t)+∑_{γ∈A}b(γ)φ̂(γ)，其中φ̂表示傅里叶变换，a是定义在ℝ的子集{λ_n}_{n≥0}上、取值为复数且具有有限指数增长的函数，ν是ℝ上的博雷尔测度，η=∑_{γ∈A}b(γ)δ_γ是支撑在该带形上的离散测度，ν和η均具有多项式增长。我们证明这些求和公式与上半平面上的殆周期亚纯Nevanlinna函数存在对应关系，更准确地说，我们根据这类函数的奇点和边界行为对上述求和公式进行了刻画，反之还证明该类中的每个函数都能生成一个上述类型的唯一傅里叶求和公式。
英文摘要
We classify Fourier summation identities in which the measure on the Fourier side is supported in a horizontal strip of $\mathbb{C}$. Let $μ=ν+η$, where $ν$ is a strongly tempered measure on $\mathbb{R}$, $η=\sum_{m\geq1} b(γ_m)δ_{γ_m}$ is a strongly tempered pure point measure supported off the real line, and $a:Λ\rightarrow\mathbb{C}$, with $Λ=\{λ_n\}_{n\geq1}\subset\mathbb{R}$, has finite exponential growth. Under natural real-antipodal and conjugation-symmetry assumptions, we characterize summation identities of the form \begin{align} \sum_{n\geq1} a(λ_n)φ(λ_n)=\int_{\mathbb{R}} \widehatφ(t)\mathrm{d}ν(t)+\sum_{m\geq1} b(γ_m)\widehatφ(γ_m), \end{align} valid for every $φ\in C^\infty_c(\mathbb{R})$, where $\widehatφ$ denotes the Fourier transform. We prove that each such identity determines a unique generating function $F$ that is holomorphic and almost periodic in the half-plane above the strip and admits a meromorphic continuation to $\mathbb{C}^+$. The measure $η$ encodes the poles and residues of $F$, while $ν$ describes the boundary behavior of its regular part through a generalized Nevanlinna representation, and $a$ determines its Fourier coefficients. Conversely, every function in the corresponding meromorphic class whose Fourier coefficients satisfy a local summability condition determines a unique summation identity of this form. The proof combines a strip version of the Bridge Lemma with a Cauchy-transform argument that accounts for the off-real poles. As an application, we show that the Guinand-Weil explicit formula for every member of the Selberg class, including non-self-dual members, fits into our framework, and we identify its associated generating function.
发表机构
- IMPA - Instituto de Matemática Pura e Aplicada（巴西纯粹与应用数学研究所）
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