Decomposition systems for function spaces
Let Θ := {θ I : e ∈ E, I ∈ D} be a decomposition system for L2(R) indexed over D, the set of dyadic cubes in R, and a finite set E, and let Θ̃ := {θ̃ I : e ∈ E, I ∈ D} be the corresponding dual functionals. That is, for every f ∈ L2(R), f = ∑ e∈E ∑ I∈D〈f, θ̃ e I〉θ I . 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, I ∈ D. Typical examples of such decomposition systems are various wavelet-type unconditional bases for L2(R), and more general systems such as affine frames.