Gaussian-derivative expansion of detector resolution functions via convolution

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, $f=G\ast h$, 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.