The author considers some problems which are potentially relevant for the construction of adaptive numerical schemes. First, he uses biorthogonal spline wavelets on \([0,1]\) to characterize the functions in Besov spaces \(B^\sigma_{r,r}(0,1)\) with \(0< \sigma <\infty\) and \((1+\sigma)^{-1} < r < \infty\). Then, the author proves results about approximation spaces with respect to restricted approximation procedures as well as that about various tree-type and tresholding approximation procedures. Also a result about \(L_\infty\)-convergence is given. Finally, singularity functions which typically arise in boundary integral equations on polygonal boundaries are investigated.