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 x V and S x T affect the frame properties of {u(-1/2) f(x/u - v)(u is an element of U,v is an element of V) and {s(-1/2) g(x/z - y)(s is an element of S,t is an element of T). In particular, it is shown that there is a relationship between the densities of he dilation sets U and S and weighted admissibility constants of f and g. This relationship implies a comparison theorem, whereby the affine densities of U x V and S x T are proportional, with proportionality constant depending on the frame bounds and the admissibility constants of f and g. These results are also extended to wavelet frame sequences. (C) 2008 Elsevier Inc. All rights reserved.