Spline wavelet decomposition in weighted function spaces

Battle-Lemarie wavelet systems of natural orders are established in the paper. The main result of the work is decomposition theorem in Besov and Triebel-Lizorkin spaces with local Muckenhoupt weights, which is performed in terms of bases generated by the systems of such a type. Battle-Lemarie wavelets are splines and suit very well for the study of integration operators.