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

TL;DR

This paper provides explicit formulas for norms on trace spaces of Sobolev spaces on real lines and constructs near-optimal interpolating splines, advancing understanding of function traces and interpolation.

Contribution

It introduces explicit expressions for trace space norms and constructs low-degree splines with near-optimal norms, linking interpolation in Sobolev spaces.

Findings

Explicit trace space norm formulas for Sobolev spaces.

Construction of low-degree interpolating splines with optimal norm bounds.

A general relation between interpolation in $L^r_p(R)$ and $W^r_p(R)$.

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 $L^r_p(R)$ and $W^r_p(R)$ for all $r \geq 1$ is also given.