Homogeneous Approximation Property

The Homogeneous Approximation Property (HAP) is a key property of Gabor systems which not only leads to interesting approximation properties but also to necessary conditions for Gabor frames in terms of the Beurling density of the associated sequence of time-frequency indices. We show that, under some mild regularity assumptions, wavelet frames also satisfy an analogue of the HAP with respect to the affine group and that this leads to necessary conditions for existence in terms of the affine density. In so doing, essential differences from the Gabor case are also revealed: we see in the wavelet case how the density is strongly tied to the generator of the frame, and there is no Nyquist density.