Decomposition systems for function spaces

Let circle minus := {theta(I)(e) : e is an element of E, I is an element of 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 circle minus := {theta(I)(e) : e is an element of E, I is an element of D} be the corresponding dual functionals. That is, for every f is an element of L-2(R-d), f = Sigma(eis an element ofE) Sigma(Iis an element ofD) <f, theta(I)(e)>theta(I)(e). We study sufficient conditions on circle minus, circle minus 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, theta(I)(e)>, e is an element of E, I is an element of 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.