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Abstract. By a general Franklin system corresponding to a dense sequence T = (t n , n ≥ 0) of points in [0, 1] we mean a sequence of orthonormal piecewise linear functions with knots T , that is, the nth function of the system has knots t 0 , . . . , t n . The main result of this paper is that each general Franklin system is an unconditional basis in L p [0, 1], 1 < p < ∞.

Abstract. We study nonlinear m-term approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basis H in L p ([0, 1]), 1 < p < ∞) a greedy type algorithm realizes nearly best m-term approximation for any individual function. In this paper we generalize this result in two directions. First, instead of a greedy algorithm we consider a weak greedy algorithm. Second, we study in detail unconditional nongreedy bases (like the multivariate Haar basis. We prove some convergence results and also some results on convergence rate of weak type greedy algorithms. Our results are expressed in terms of properties of the basis with respect to a given weakness sequence.