AI 中文总结
本文提出傅里叶空间框架，将高斯与核函数卷积的分辨率函数展开为高斯导数级数，建立可容许性准则并分类多种实际核函数，为非高斯分辨率参数化提供统一比较与构建语言。
AI 中文摘要
在基于探测器的谱学和高能物理领域，窄共振的观测线形会因探测器的有限分辨率发生畸变。尽管高斯函数是标准的一阶近似，但实际分辨率函数会以非平凡的方式偏离高斯分布。本文提出一种统一的傅里叶空间框架，其中可将可写为高斯函数与核函数卷积形式的分辨率函数（即$f=G*h$）展开为高斯导数级数。以Voigt分布作为原型，该机制可扩展至傅里叶变换在原点处解析且为指数型的任意核函数。本文建立了一个可容许性准则，随后检验了几种实际应用中的核函数——弥散Hypatia、Cruijff、拉普拉斯分布及广义双曲核——将其分别归类为直接可容许、微扰可容许或渐近可容许。该框架因此为围绕高斯函数构建和比较非高斯分辨率参数化提供了统一语言。
英文摘要
In detector-based spectroscopy and high-energy physics, the observed lineshape of a narrow resonance is distorted by the finite resolution of the detector. While the Gaussian is the standard first approximation, realistic resolution functions deviate from it in nontrivial ways. We present a unified Fourier-space framework in which a resolution function that can be written as a Gaussian convolved with a kernel, $f=G*h$, is expanded into a series of Gaussian derivatives. The Voigtian distribution serves as the prototype, and the mechanism extends to arbitrary kernels whose Fourier transforms are analytic at the origin and of exponential type. We establish an admissibility criterion, and then examine several kernels used in practice--- smeared Hypatia, Cruijff, the Laplacian distribution, and the generalised hyperbolic core---classifying each as directly, perturbatively, or asymptotically admissible. The framework thus provides a common language for comparing and constructing non-Gaussian resolution parametrisations around the Gaussian.
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