Franklin system is a system of functions consisting of piecewise linear conitunuous functions forming an orthonormal basis in the space of continuous function $C[0,1]$ (this is the system resulting in Gram–Schmidt orthogonaliztion of the classical Faber-Schauder system). The paper studies convergence properties of weighted greedy algorithms with respect to Franklin system renormalized in $||\cdot||_{C[0,1]}$-norm. We characterize the all weighted greedy algorithms which converge uniformly for continuous functions and almost everywhere for integrable functions. In case, when the algorithm fails to satisfy our classification criteria, we construct a continuous function for which the corresponding approximation diverges unboundedly almost everywhere. For the construction we develop certain analogues of stopping-time tools for Franklin system which are of independent interest. The results are similar in spirit with the case of Haar system treated here, however technicalities are quite different and are more involved. Some applications of the methods developed in this paper to wavelet systems are also discussed.

Given the Fourier-Haar series of a function $f\in L^1(0,1)$, with the Haar system normalized in $L^\infty$, the paper studies convergence properties of a rearranged series of $f$ where the coefficients, multiplied by some positive weights (i.e. a decreasing sequence of positive numbers) $\Gamma=\{\gamma_n\}_{n=0}^\infty $, are taken in decreasing order of their magnitudes (weighted greedy algorithm). This method of summation of the Haar series of $f$ is non-linear (with respect to $f$) and contains, as a particular case, the so-called hard sampling method in it. In this paper we give a complete description of all weights for which such decreasing rearrangements converge uniformly for all continuous functions and almost everywhere for all integrable functions. If the weight $\Gamma$ fails to satisfy our condition, which reads $$ \sup\limits_{m>n>0} \left\{ \frac{m}{n}: \frac{\gamma_n}{\gamma_m}\leq 2 \right\} < \infty, $$ we construct a continuous function such that its weighted greedy algorithm diverges unboundedly almost everywhere. For this last construction, we use the connection of Fourier-Haar series with martingales and employ stopping-time techniques.