Sharp inverse statements for kernel interpolation
Tizian Wenzel
Abstract
While direct statements for kernel based interpolation on regions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega subset-of double-struck upper R Superscript d"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo> ⊂ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\Omega \subset \mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are well researched, far less is known about corresponding inverse statements. The available inverse statements for kernel based interpolation so far are not sharp. In this paper, we derive sharp inverse statements for interpolation using finitely smooth kernels, such as popular radial basis function kernels like the class of Matérn or Wendland kernels. In particular, the results show that there is a one-to-one correspondence between the smoothness of a function and its approximation rate via kernel interpolation: If a function can be approximated with a given rate, it has a corresponding smoothness and vice versa.