Inequalities of Littlewood–Paley Type for Frames and Wavelets

Inequalities of Littlewood–Paley type for frames in both the wavelet and Weyl–Heisenberg settings, and those for any unconditional basis of the form $psi _{j,k} (x) = 2^{frac{j}{2}} psi (2^j x - k)$, are established. In particular, if ${ psi _{j,k} } $ is a semi-orthogonal basis, then the Littlewood-Paley identity is obtained. A similar identity for the “biorthogonal wavelets” of Cohen, Daubechies, and Feauveau is also obtained.