It is well-known that isotropic Besov spaces can be characterized by wavelets. Probably less well-known are those function spaces which are characterized by tensor product wavelet systems. Specific examples are certain tensor products of Besov spaces.
We shall discuss the quality of best m-term approximation with respect to these two examples, i.e., Besov spaces and tensor products of Besov spaces. More exactly, we shall investigate the asymptotic behaviour of the widths of best m-term approximation, here denoted by σ_m(Y,X,Φ).
In our lectures, usually X = Lp(Ω), Φ denotes the wavelet system and Y will be either an isotropic Besov space or a tensor product Besov space. Our approach relies on the abstract theory of approximation spaces which we will shortly recall. Methods from interpolation theory (in particular real interpolation) will also play a role. We shall spend some time to discuss the similarities as well as the differences between the isotropic and the tensor product situation.
Finally, we shall compare best m-term approximation with optimal linear approximation. For this purpose we shall also introduce and discuss a few properties of approximation numbers (linear widths).