## Approximation Numbers of Sobolev Embeddings

Let denote the Sobolev space of -times
differentiable functions in on a bounded
Lipschitz domain in
.
If and
,

with
and
the symbol indicating inequalities in both
directions with constants not depending on
.
Asymptotically optimal approximation operators
improve the exponent for standard schemes,
such as spline interpolants or finite element
projections, by .
The gain is maximal for , , when
.

The figure shows the exponents for all values of
.
Apparently, classical methods are optimal for and
(white area).

Dissertation, Bonn, 1979

Math. Annalen, 242 (1979), 273-281