Comparison theorems for separable wavelet frames
The author studies the relationships between the affine densities of the sets \(U\times V\) and \(S\times T\) (both subsets of \(\subset (0,\infty)\times (-\infty,\infty)\)) and the frame properties of \(\{u^{-1/2}f(\frac xu-v)\}_{u\in U,v\in V}\) and \(\{s^{-1/2}f(\frac xs-t)\}_{s\in U,t\in V}\) and in particular it is shown that the quotient of the densities of the dilation sets \(U\) and \(S\) can be bounded from above and below by terms involving the admissibility constants \(C_f= \int_{-\infty}^\infty |\hat f(w)|^2\frac{dw}{|w|}\), \(C_g\) and frame bounds. A similar proportionality result then follows fro the affine densities of \(U\times V\) and \(S\times T\). These results are extended to wavelet frame sequences as well.
scientific article; zbMATH DE number 5655029
scientific article