Title: Schoenberg Representations and Gramian Matrices of Matérn Functions,
Abstract: We represent Matérn functions in terms of Schoenberg's integrals which
ensure the positive definiteness and prove the systems of translates of
Matérn functions form Riesz sequences in $L^2(\R^n)$ or Sobolev spaces. Our
approach is based on a new class of integral transforms that generalize Fourier
transforms for radial functions. We also consider inverse multi-quadrics and
obtain similar results.