Optimal approximation of elliptic problems by linear and nonlinear mappings III: Frames

We study the optimal approximation of the solution of an operator equation A(u)=f by certain n-term approximations with respect to specific classes of frames. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space and Ω⊂Rd is a bounded Lipschitz domain; the error is always measured in the Hs-norm. We study the order of convergence of the corresponding nonlinear frame widths and compare it with several other approximation schemes. Our main result is that the approximation order is the same as for the nonlinear widths associated with Riesz bases, the Gelfand widths, and the manifold widths. This order is better than the order of the linear widths iff p<2. The main advantage of frames compared to Riesz bases, which were studied in our earlier papers, is the fact that we can now handle arbitrary bounded Lipschitz domains—also for the upper bounds.