In this paper we present applications of methods from wavelet analysis to polynomial approximations for a number of nonlinear problems. In the general case we have the solution as a multiresolution expansion based on compactly supported wavelet. The solution is parametrized by solutions of two reduced algebraical problems, one is nonlinear and the other is linear problem, which is obtained from one of the next wavelet constructions: fast wavelet transform, stationary subdivision schemes, and the method of connection coefficients.
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