Sobolev Duals in Frame Theory and Sigma-Delta Quantization

A new class of alternative dual frames is introduced in the setting of finite frames for ? ^d . These dual frames, called Sobolev duals, provide a high precision linear reconstruction procedure for Sigma-Delta (ΣΔ) quantization of finite frames. The main result is summarized as follows: reconstruction with Sobolev duals enables stable rth order Sigma-Delta schemes to achieve deterministic approximation error of order $\mathcal{O}(N^{-r})$ for a wide class of finite frames of size N. This asymptotic order is generally not achievable with canonical dual frames. Moreover, Sobolev dual reconstruction leads to minimal mean squared error under the classical white noise assumption.