General Franklin systems as bases in H¹[0,1]

Abstract

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].

On absolute convergence of series by a general Franklin system

“…If δ is to the left of Δ n , then (1.6) follows from (1.1) and (2.1). In the case where δ is to the right of Δ n , using the equality The next lemma was proved in [3], Lemma 3.4.…”

“…To study the properties of general Franklin systems, in [2] and [3] were introduced the notions of strong and weak regularity of a partition T . The corresponding definitions follow.…”

Общая Система Франклина Как Базис В Пространстве $B^1[0,1]$

“…Очевидно, что если последовательность сильно регулярная или сильно регулярная по парам, то |J n | ∼ γ |∆ n |. Поэтому (ради удобства) в оценках, полученных в работах [12], [9], [8], для функций f n мы можем J n заменить на ∆ n .…”

“…Доказательство. Здесь мы в основном следуем схеме доказательства леммы 5.2 из работы [8]. Многие обозначения повторяют обозначения из доказательства леммы 6.…”

Ciesielski and Franklin systems

“…In [15,16,17] several types of regularities of partitions were defined. Let T = {t i : i ≥ 0} be an admissible sequence of points and let σ n = {0 = t n,0 < t n,1 ≤ .…”