In this note we will show that the so called Sobolev dual is the minimizer over all linear reconstructions using dual frames for stable <i>r<sup>th</sup></i> order ΣΔ quantization schemes under the so called White Noise Hypothesis (WNH) design criteria. We compute some Sobolev duals for common frames and apply them to audio clips to test their performance against canonical duals and another alternate dual corresponding to the well known Blackman filter.
The theory of localized frames is a recently introduced concept with
broad implications to frame theory in general, as well as to the
special cases of Gabor and wavelet frames. Using the new notion of a
R-dual sequence associated with a Bessel sequence, we derive
several duality principles concerning localization in abstract frame
theory. As applications of our results we prove a duality principle
of localization of Gabor systems in the spirit of the Ron-Shen
duality principle, and obtain a Janssen representation for general
We examine the question of which characteristic functions yield Weyl-Heisenberg frames for various values of the parameters. We also give numerous applications of frames of characteristic functions to the general case (g, a, b).