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

## TL;DR

The paper introduces three classes of compactly supported functions that serve as reproducing kernels for Sobolev spaces of any order, using novel oscillatory integral transforms involving Fourier and Hankel transforms.

## Contribution

It presents a new method for constructing reproducing kernels for Sobolev spaces using oscillatory integral transforms with Fourier and Hankel transforms.

## Key findings

- Three classes of compactly supported reproducing kernels are constructed.
- The method applies to Sobolev spaces of arbitrary order greater than d/2.
- The approach involves innovative oscillatory integral transforms combining Fourier and Hankel 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.