On general Franklin systems

Gevorkyan Gegham ; Kamont Anna

GDML_Books, (1998), p.

Résumé

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 $L^p$, 1 < p < ∞, and $H^p$, 1/2 < p ≤ 1; equivalent conditions for the unconditional convergence of the Franklin series in $L^p$ for 0< p ≤ 1; relation between Haar and Franklin series with identical coefficients; characterization of the spaces BMO and Lip(α), 0 < α < 1, in terms of the Fourier-Franklin coefficients.

CONTENTS1. Introduction.....................................................................................5 1.1. Notation......................................................................................72. Definition and properties of general Franklin systems..................10 2.1. Piecewise linear functions.........................................................10 2.2. Franklin functions.....................................................................11 2.3. Sequences of partitions and Franklin functions........................13  2.3.1. Regularity of sequences of partitions...................................14 2.4. Sequences of partitions and general Haar systems.................16 2.5. Technical lemmas.....................................................................173. Franklin series in $L^p$, 1 < p < ∞...............................................214. Franklin series in $L^p$, 0 < p ≤ 1, and $H^p$, 1/2 < p ≤ 1..........275. The necessity of strong regularity in $H^p$, 1/2 < p ≤ 1...............426. Haar and Franklin series with identical coefficients......................467. Characterization of the spaces BMO and Lip(α), 0 < α < 1...........51References.......................................................................................58

1991 Mathematics Subject Classification: Primary 42C10.

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Zbl 0910.42016

EUDML-ID : urn:eudml:doc:271242