Paley-Wiener Theorems For Slice Regular Functions
Yanshuai Hao, Pei Dang, Weixiong Mai
TL;DR
This work extends Paley-Wiener theory to slice regular quaternion-valued functions by establishing two parallel forms: a compact-type theorem that equates exponential-type growth with uniform spectral support $\operatorname{supp}\mathscr{F}_{I}(f|_{\mathbb{R}})\subset [-A,A]$ for all imaginary units $I$, and a non-compact-type theorem grounded in Hardy spaces $H^{p}(\mathbb{H}_{+})$ with $1\le p\le 2$ and spectral support $\operatorname{supp}\mathscr{F}_{E}(f)\subset(-\infty,0]$. The compact-type result yields an explicit slice-wise representation $f(x+Iy)=\frac{1}{\sqrt{2\pi}}\int_{-A}^{A} e^{I(x+Iy)t}\mathscr{F}_{I}(f|_{\mathbb{R}})(t)\,dt$, and the authors develop the reproducing kernel for the slice-regular Paley-Wiener space along with a corresponding sampling theorem. In the non-compact case, the theorems provide necessary and sufficient conditions linking boundary values and the essential Fourier transform, enabling construction of functions in $H^{p}(\mathbb{H}_{+})$ from data on the real line. The results culminate in a quaternionic Paley-Wiener framework that supports stable reconstruction and sampling via a quaternionic sinc kernel, extending classical Fourier-analytic Paley-Wiener theory to slice-regular function theory.
Abstract
We prove two theorems of Paley and Wiener in the slice regular setting. As an application, we can compute the reproducing kernel for the slice regular Paley-Wiener space, and obtain a related sampling theorem.