Multiwavelet sampling theorem in Sobolev spaces

Abstract

This paper is to establish the multiwavelet sampling theorem in Sobolev spaces. Sampling theorem plays a very important role in digital signal communication. The most classical sampling theorem is Shannon sampling theorem, which works for bandlimited signals. Recently, sampling theorems in wavelets or multiwavelets subspaces are extensively studied in the literature. In this paper, we firstly propose the concept of dual multiwavelet frames in dual Sobolev spaces (Hs(ℝ), H−s(ℝ)). Then we construct a special class of dual multiwavelet frames, from which the multiwavelet sampling theorem in Sobolev spaces is obtained. That is, for any f ∈ Hs(ℝ) with s > 1/2, it can be exactly recovered by its samples. Especially, the sampling theorem works for continuous signals in L2(ℝ), whose Sobolev exponents are greater than 1/2.