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.
We give a simple geometric characterization of knot sequences for which the corresponding orthonormal spline system of arbitrary order k is an unconditional basis in the atomic Hardy space H¹[0,1].
AbstractWe study general Franklin systems, i.e. systems of orthonormal piecewise linear functions corresponding to quasi-dyadic sequences of partitions of [0,1]. The following problems are treated: unconditionality of the general Franklin basis in , 1 < p < ∞, and , 1/2 < p ≤ 1; equivalent conditions for the unconditional convergence of the Franklin series in for 0< p ≤ 1; relation between Haar and Franklin series with identical coefficients; characterization of the spaces BMO and...
We investigate when the trigonometric conjugate to the periodic general Franklin system is a basis in C(𝕋). For this, we find some necessary and some sufficient conditions.
By a general Franklin system corresponding to a dense sequence of knots 𝓣 = (tₙ, 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 t₀, ..., tₙ. 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 H¹[0,1].
By a general Franklin system corresponding to a dense sequence = (tₙ, n ≥ 0) of points in [0,1] we mean a sequence of orthonormal piecewise linear functions with knots , that is, the nth function of the system has knots t₀, ..., tₙ. The main result of this paper is that each general Franklin system is an unconditional basis in , 1 < p < ∞.
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