BMO, boundedness of affine operators, and frames

This paper addresses the construction of wavelet frames as an application of the modern theory of singular integrals. The continuous wavelet inversion formula (Calderón reproducing formula) may be viewed as the action of a Calderón-Zygmund singular integral operator. Wavelet frame operators arise as Riemann sum approximations of these singular integrals. When the analyzing and synthesizing functions are smooth and have a vanishing moment, boundedness of the approximations is a simple matter of applying, for example, the Cotlar lemma. Here we investigate the situation when only one of the analyzing/synthesizing pair has a vanishing moment. The dyadic discretizations are no longer automatically bounded. We show how the T(1) theorem may be used to find criteria under which boundedness and invertibility are ensured. Parallels between these ideas and the frame criteria of Daubechies and Ron-Shen are also discussed.