Density of weighted wavelet frames

Abstract

If ψ ∈ L²(R), Λ is a discrete subset of the affine groupA =R ⁺ ×R, and w: Λ →R ⁺ is a weight function, then the weighted wavelet system generated by ψ, Λ, and w is$\mathcal{W}(\psi ,\Lambda ,\omega ) = \{ \omega (a,b)^{1/2} a^{ - 1/2} \psi (\frac{x}{a} - b):(a,b) \in \Lambda \}$. In this article we define lower and upper weighted densities D ⁻_w (Λ) and D ⁺_w (Λ) of Λ with respect to the geometry of the affine group, and prove that there exist necessary conditions on a weighted wavelet system in order that it possesses frame bounds. Specifically, we prove that if W(ψ, Λ, w) possesses an upper frame bound, then the upper weighted density is finite. Furthermore, for the unweighted case w = 1, we prove that if W(ψ, Λ, 1) possesses a lower frame bound and D ⁺_w (Λ⁻¹) < ∞, then the lower density is strictly positive. We apply these results to oversampled affine systems (which include the classical affine and the quasi-affine systems as special cases), to co-affine wavelet systems, and to systems consisting only of dilations, obtaining some new results relating density to the frame properties of these systems.