STUDIA MATHEMATICA 157 (2) (2003) Decomposition systems for function spaces by G. Kyriazis (Nicosia) Abstract. Let Θ := {θ e I : e ∈ E, I ∈ D} be a decomposition system for L 2 (R d ) indexed over D, the set of dyadic cubes in R d , and a finite set E, and let  Θ := {  θ e I : e ∈ E, I ∈ D} be the corresponding dual functionals. That is, for every f ∈ L 2 (R d ), f = ∑ e∈E ∑ I ∈D 〈f,  θ e I 〉θ e I . We study sufficient conditions on Θ,  Θ 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,  θ e I 〉, e ∈ E, I ∈ D. Typical examples of such decomposition systems are various wavelet-type unconditional bases for L 2 (R d ), and more general systems such as affine frames. 1. Introduction. Let E be a finite set and D be the family of dyadic cubes in R d . Given a decomposition system Θ := {θ e I : e ∈ E, I ∈ D}, for L 2 (R d ) with dual functionals  Θ := {  θ e I : e ∈ E, I ∈ D} our goal is to study sufficient conditions on Θ,  Θ so that they constitute a decomposition system for the homogeneous Triebel–Lizorkin and Besov spaces. That is, every distribution f in the above spaces can be expressed in the form f =  e∈E  I ∈D 〈f,  θ e I 〉θ e I . Moreover, we are interested in characterizing the membership of a distribu- tion f in the Triebel–Lizorkin and Besov spaces by the size of the coefficients 〈f,  θ e I 〉, e ∈ E, I ∈ D. Typical examples of such systems are various uncon- ditional bases for L 2 (R d ) such as the biorthogonal wavelet bases, the bases constructed in [Pet] or even the affine frames of L 2 (R d ). To describe our results we first introduce the standard multi-index no- tation. For x =(x 1 ,...,x d ) ∈ R d and α =(α 1 ,...,α d ) ∈ N d (N := {n : n ≥ 0}, d ≥ 1), we let |x| := √ x 2 1 + ... + x 2 d , x α := x α 1 1 ...x α d d , |α| := α 1 + ... + α d , α!= α 1 ! ...α d !, and (·) (α) := ∂ |α| (·)/∂x α 1 1 ...∂x α d d . 2000 Mathematics Subject Classification : 41A17, 41A20, 42B25, 42C15. Key words and phrases : unconditional bases, wavelets, frames, Besov spaces, Triebel– Lizorkin spaces. [133]