New bases for Triebel-Lizorkin and Besov spaces (Q2759075)

The authors give a new method for the construction of unconditional bases for general classes of Triebel-Lizorkin and Besov spaces. These include the \(L_p\), \(H_p\), potential and Sobolev spaces. The main feature of their method is that the character of the basis function can be prescribed in a very general way. In particular, if \(\Phi\) is any sufficiently smooth and rapidly decaying function then their method constructs a basis whose elements are linear combinations of a fixed (small) number of shifts and dilates of the single function \(\Phi\). Typical examples of such \(\Phi\)'s are the rational functions \(\Phi(\cdot)= (1+|\cdot |^2)^{-N}\) and the Gaussian function \(\Phi(\cdot)=e^{-|\cdot |^2}\). The paper also shows how the new bases can be utilized in nonlinear approximation.