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

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