Wavelet-Galerkin approximation of linear translation invariant operators

It is shown that the wavelet-Galerkin discretization of linear translation invariant (LTI) operators has good numerical properties, arising from the vanishing moments property of wavelets. If a wavelet has M vanishing moments, then it can have at most M-1 continuous derivatives, and hence operators of the form d^p /dx^p, where p>M, have to be considered as generalized derivatives. Even in this case the approximation results derived hold. Also, the virtual expansion theorem is useful in the sense that there is no need to compute the expansion coefficients of the function at some level V_▵x