Compactly supported reproducing kernels for $L^2$-based Sobolev spaces and Hankel-Schoenberg transforms

We exhibit three classes of compactly supported functions which provide reproducing kernels for the Sobolev spaces $H^{\delta }({\mathbb{R}}^d)$ of arbitrary order $\delta >d\mathrm{/}2.$ Our method of construction is based on a new class of oscillatory integral transforms that incorporate radial Fourier transforms and Hankel transforms.