Generalized sampling theorems in multiresolution subspaces

Abstract

It is well known that under very mild conditions on the scaling function, multiresolution subspaces are reproducing kernel Hilbert spaces (RKHSs). This allows for the development of a sampling theory. In this paper, we extend the existing sampling theory for wavelet subspaces in several directions. We consider periodically nonuniform sampling, sampling of a function and its derivatives, oversampling, multiband sampling, and reconstruction from local averages. All these problems are treated in a unified way using the perfect reconstruction (PR) filter bank theory. We give conditions for stable reconstructions in each of these cases. Sampling theorems developed in the past do not allow the scaling function and the synthesizing function to be both compactly supported, except in trivial cases. This restriction no longer applies for the generalizations we study here, due to the existence of FIR PR banks. In fact, with nonuniform sampling, oversampling, and reconstruction from local averages, we can guarantee compactly supported synthesizing functions. Moreover, local averaging schemes have additional nice properties (robustness to the input noise and compression capabilities). We also show that some of the proposed methods can be used for efficient computation of inner products in multiresolution analysis. After this, we extend the sampling theory to random processes. We require autocorrelation functions to belong to some subspace related to wavelet subspaces. It turns out that we cannot recover random processes themselves (unless they are bandlimited) but only their power spectral density functions. We consider both uniform and nonuniform sampling.