Comparison theorems for separable wavelet frames

It is known that wavelet frames do not exhibit a Nyquist density. Even so, this paper shows that the affine densities of the sets U×VU×V and S×TS×T affect the frame properties of {u−12f(xu−v)}u∈U,v∈V and {s−12g(xs−t)}s∈S,t∈T. In particular, it is shown that there is a relationship between the densities of the dilation sets UU and SS and weighted admissibility constants of ff and gg. This relationship implies a comparison theorem, whereby the affine densities of U×VU×V and S×TS×T are proportional, with proportionality constant depending on the frame bounds and the admissibility constants of ff and gg. These results are also extended to wavelet frame sequences.