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

$\displaystyle 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}$.

EPSRC Gregynog Workshop, 21-26 July 2002

Gregynog Abstracts