Peter Oswald

Bell Labs

Lucent Technologies

**Abstract:**
I will give some examples of nonlinear approximation problems
arising in connection with adaptive algorithms for operator
equations as well as for image and signal processing. Many
of them can be cast into the form of nonlinear *N*-term
approximation: For a given ``function'' *f*, find a linear combination
of at most *N* functions from a given ``dictionary'' such that a certain
error measure is optimized. For larger, redundant dictionaries, there
are a lot of open theoretical and algorithmical problems associated
with *N*-term approximation.
What is well-understood is a case of multivariate *N*-term
approximation from B-spline and wavelet dictionaries in function
spaces that models isotropic h-refinement in the finite element
method. I show another example that analyzes also certain anisotropic
refinement.