Fourier Analysis of Petrov-Galerkin Methods Based on Biorthogonal Multiresolution Analyses

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When solving differential equations by means of a Galerkin approach, the approximating spaces are not only supposed to have good approximation properties, but also they must allow easy and fast computations. In addition, if the goal is the development of a multilevel method to detect and follow local singularities, or a multigrid scheme to solve the resulting linear systems, then hierarchical bases are necessary. As an example, we mention the finite element bases which have been widely used over the last three decades.