Explicit traces of functions on Sobolev spaces and quasi-optimal linear interpolators

Abstract

Let $\Lambda \subset R$ be a strictly increasing sequence. For $r=1,2$, we give a simple explicit expression for an equivalent norm on the trace spaces $W_p^r(R){|}_{\Lambda }$, $L_p^r(R){|}_{\Lambda }$ of the non-homogeneous and homogeneous Sobolev spaces with $r$ derivatives $W_p^r(R)$, $L_p^r(R)$. We also construct an interpolating spline of low degree having optimal norm up to a constant factor. A general result relating interpolation in $L_p^r(R)$ and $W_p^r(R)$ for all $r\ge 1$ is also given.

arXiv comment12 pages; 1 figure; Minor update regarding the dependence of constants on the parameter p