Wavelets with differential relation (Q546353)

Divergence-free wavelets are useful in wavelet-based numerical solutions of Navier-Stokes equations and in the analysis of incompressible flows. A differential relation is required in divergence-free wavelets, more precisely, \[ \varphi_1'(x)=\sum_{k} c_k \varphi_0(x-k),\qquad \psi_1'(x)=\sum_{k} d_k \psi_0(x-k), \] where \((\varphi_0, \psi_0)\) and \((\varphi_1, \psi_1)\) are two groups of scaling functions and wavelets. This paper points out some restrictions of these wavelets with a differential relation, for example, it has been shown in Theorem~2.1 that both \(\varphi_0\) and \(\varphi_1\) cannot simultaneously have the interpolatory property. More generally, biorthogonal wavelets with the differential relation in (4.1)--(4.4) have been studied in Section~4. Quite a few examples, including many interpolatory wavelets, have been provided in the paper to illustrate the differential relation and divergence-free wavelets.