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.
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A small perturbation method is developed and employed to construct frames with compactly supported elements of small shrinking support for Besov and Triebel-Lizorkin spaces in the general setting of a doubling metric measure space in the presence of a nonnegative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. This allows one, in particular, to construct compactly supported frames for Besov and Triebel-Lizorkin spaces on the sphere, on the interval with Jacobi weights as well as on Lie groups, Riemannian manifolds, and in various other settings. The compactly supported frames are utilized to introduce atomic Hardy spaces $H^{p}_{A}$ in the general setting of this article.
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