On quantization of finite frame expansions:: σ-δ schemes of arbitrary order

In this note we will show that the so called Sobolev dual is the minimizer over all linear reconstructions using dual frames for stable r(th) order Sigma Delta 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.