On general Franklin systems

Gevorkyan Gegham, and Kamont Anna. On general Franklin systems. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1998. <http://eudml.org/doc/271242>.

@book{GevorkyanGegham1998,
abstract = {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.......................................................................................581991 Mathematics Subject Classification: Primary 42C10.},
author = {Gevorkyan Gegham, Kamont Anna},
keywords = {Franklin system; unconditional basis; unconditional convergence; $L^p$ spaces; real Hardy spaces; Hölder classes; BMO space; dyadic Hardy spaces; BMO; Franklin-Fourier coefficients},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {On general Franklin systems},
url = {http://eudml.org/doc/271242},
year = {1998},
}

TY - BOOK
AU - Gevorkyan Gegham
AU - Kamont Anna
TI - On general Franklin systems
PY - 1998
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - 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.......................................................................................581991 Mathematics Subject Classification: Primary 42C10.
LA - eng
KW - Franklin system; unconditional basis; unconditional convergence; $L^p$ spaces; real Hardy spaces; Hölder classes; BMO space; dyadic Hardy spaces; BMO; Franklin-Fourier coefficients
UR - http://eudml.org/doc/271242
ER -