Decomposition systems for function spaces

Abstract

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