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. By a general Franklin system corresponding to a dense sequence of knots T = (t n , n ≥ 0) 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 a characterization of sequences T for which the corresponding general Franklin system is a basis or an unconditional basis in H 1 [0, 1].
A short survey of results on classical Franklin system, Ciesielski systems and general Franklin systems is given. The principal role of the investigations of Z. Ciesielski in the development of these three topics is presented. Recent results on general Franklin systems are discussed in more detail. Some open problems are posed.
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