PRECISE BEHAVIOUR OF THE APPROXIMATION NUMBERS OF THE SOBOLEV EMBEDDING IN THE ONE-DIMENSION CASE

JAN LANG

langjan@yahoo.com, lang@math.ohio-state.edu

Department of Mathematics
The Ohio State University
231 W 18th Avenue
Columbus, OH 43210, USA

N We consider the Sobolev embedding $I:W^{1,p}(a,b) \to L^p(a,b),$ $-\infty < a < b < \infty$ and $1<p<\infty$. We show that for all $n$ in ${\bf N}$ the $n$-th approximation number $a_n(I)$ of $I$ is given by

$$a_n(I)={c_p \over n}$$

where $c_p$ is a constant dependent only on $p$. We also find the best $n$ dimensional linear approximation of $I$ and we will discuss the shape of unit ball in $W^{1,p}$.