Unconditionality of general Franklin systems in L^p[0,1], 1<p<∞

Abstract

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 < ∞.

Unconditionality of periodic orthonormal spline systems in $L^p$

“…the case k = 2) forms an unconditional basis in L p [0, 1], 1 < p < ∞. Here we obtain an estimate for general periodic orthonormal spline functions, which combined with the methods developed in [10] yield the unconditionality of periodic orthonormal spline systems in L p (T).…”

“…Considerable effort has been made in the past to weaken the restriction to dyadic knot sequences. In the series of papers [9,11,10] this restriction was removed step-by-step for general Franklin systems, with the final result that it was shown for each admissible point sequence (t n ) n≥0 with parameter k = 2, the associated general Franklin system forms an unconditional basis in L p [0, 1], 1 < p < ∞. Combining the methods used in [11,10] with some new inequalities from [15] it was proved in [13] that non-periodic orthonormal spline systems are unconditional bases in L p [0, 1], 1 < p < ∞, for any spline order k and any admissible point sequence (t n ).…”

On absolute convergence of series by a general Franklin system

“…It is clear that if the sequence T is weakly regular, then |J n | ∼ γ |Δ n |. Therefore, in the estimates obtained in papers [2] and [5] for function f n we can replace J n by Δ n . Now we list some estimates for Franklin function, obtained in [2] and [5].…”

“…Therefore, in the estimates obtained in papers [2] and [5] for function f n we can replace J n by Δ n . Now we list some estimates for Franklin function, obtained in [2] and [5]. First, there exists a constant C γ , such that for all n and x…”

“…In [5] it was proved that there exist constants c and C, such that for any p (1 ≤ p ≤ ∞) the following chain of inequalities holds:…”