### On bases and unconditional bases in the spaces ${L}^{p}\left(d\mu \right)$,1≤p<∞

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We study sufficient conditions on the weight w, in terms of membership in the ${A}_{p}$ classes, for the spline wavelet systems to be unconditional bases of the weighted space ${H}^{p}\left(w\right)$. The main tool to obtain these results is a very simple theory of regular Calderón-Zygmund operators.

Properties of representation systems with respect to summation methods are studied. For a given representation system with respect to a given summation method we study, in particular, the question of the stability of that property after deleting finitely many elements. As a consequence we obtain the existence of null series for the systems with respect to a given method of summation.

The aim of this paper is to obtain sharp estimates from below of the measure of the set of divergence of the m-fold Fourier series with respect to uniformly bounded orthonormal systems for the so-called G-convergence and λ-restricted convergence. We continue the study begun in a previous work.

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