Decomposition systems for function spaces

G. Kyriazis. "Decomposition systems for function spaces." Studia Mathematica 157.2 (2003): 133-169. <http://eudml.org/doc/285306>.

@article{G2003,
abstract = {Let $Θ: = \{θ_\{I\}^\{e\}: e ∈ E, I ∈ D\}$ be a decomposition system for $L₂(ℝ^\{d\})$ indexed over D, the set of dyadic cubes in $ℝ^\{d\}$, and a finite set E, and let $Θ̃: = \{Θ̃ _\{I\}^\{e\}: e ∈ E, I ∈ D\}$ be the corresponding dual functionals. That is, for every $f ∈ L₂(ℝ^\{d\})$, $f = ∑_\{e∈E\} ∑_\{I∈D\} ⟨f,Θ̃_\{I\}^\{e\}⟩ θ_\{I\}^\{e\}$. We study sufficient conditions on Θ,Θ̃ so that they constitute a decomposition system for Triebel-Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution f in these spaces by the size of the coefficients $⟨f,Θ̃_\{I\}^\{e\}⟩$, e ∈ E, I ∈ D. Typical examples of such decomposition systems are various wavelet-type unconditional bases for $L₂(ℝ^\{d\})$, and more general systems such as affine frames.},
author = {G. Kyriazis},
journal = {Studia Mathematica},
keywords = {decomposition system; wavelets; frames; Besov spaces; Triebel-Lizorkin spaces; biorthogonal wavelet bases; unconditional bases},
language = {eng},
number = {2},
pages = {133-169},
title = {Decomposition systems for function spaces},
url = {http://eudml.org/doc/285306},
volume = {157},
year = {2003},
}

TY - JOUR
AU - G. Kyriazis
TI - Decomposition systems for function spaces
JO - Studia Mathematica
PY - 2003
VL - 157
IS - 2
SP - 133
EP - 169
AB - Let $Θ: = {θ_{I}^{e}: e ∈ E, I ∈ D}$ be a decomposition system for $L₂(ℝ^{d})$ indexed over D, the set of dyadic cubes in $ℝ^{d}$, and a finite set E, and let $Θ̃: = {Θ̃ _{I}^{e}: e ∈ E, I ∈ D}$ be the corresponding dual functionals. That is, for every $f ∈ L₂(ℝ^{d})$, $f = ∑_{e∈E} ∑_{I∈D} ⟨f,Θ̃_{I}^{e}⟩ θ_{I}^{e}$. We study sufficient conditions on Θ,Θ̃ so that they constitute a decomposition system for Triebel-Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution f in these spaces by the size of the coefficients $⟨f,Θ̃_{I}^{e}⟩$, e ∈ E, I ∈ D. Typical examples of such decomposition systems are various wavelet-type unconditional bases for $L₂(ℝ^{d})$, and more general systems such as affine frames.
LA - eng
KW - decomposition system; wavelets; frames; Besov spaces; Triebel-Lizorkin spaces; biorthogonal wavelet bases; unconditional bases
UR - http://eudml.org/doc/285306
ER -