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Ciesielski and Franklin systems

100%
Gevorkyan G.
Banach Center Publications
|
2006
|
tom 72
|
nr 1
85-92
EN
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.
2
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General Franklin systems as bases in H¹[0,1]

63%
Gevorkyan G., Kamont A.
Studia Mathematica
|
2005
|
tom 167
|
nr 3
259-292
EN
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].
3
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Unconditionality of general Franklin systems in $L^{p}[0,1]$, 1 < p < ∞

63%
Gevorkyan G., Kamont A.
Studia Mathematica
|
2004
|
tom 164
|
nr 2
161-204
EN
By a general Franklin system corresponding to a dense sequence 𝓣 = (tₙ, 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 t₀, ..., tₙ. The main result of this paper is that each general Franklin system is an unconditional basis in $L^{p}[0,1]$, 1 < p < ∞.
4
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On the trigonometric conjugate to the general Franklin system

63%
Gevorkyan G., Kamont A.
Studia Mathematica
|
2009
|
tom 193
|
nr 3
203-239
EN
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.
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