Root-Exponential Accuracy for Coarse Quantization of Finite Frame Expansions

In this note, we show that by quantizing the N-dimensional frame coefficients of signals in R-d using rth-order Sigma-Delta quantization schemes, it is possible to achieve root-exponential accuracy in the oversampling rate lambda := N/d. In particular, we construct a family of finite frames tailored specifically for coarse Sigma-Delta quantization that admit themselves as both canonical duals and Sobolev duals. Our construction allows for error guarantees that behave as e(-c root lambda), where under a mild restriction on the oversampling rate, the constants are absolute. Moreover, we show that harmonic frames can be used to achieve the same guarantees, but with the constants now depending on d.