Recovery of the optimal approximation from samples in wavelet subspace

The success of the typical sampling theories for a wavelet subspace mostly benefits from the fact that the sampling operation is an isomorphism of a wavelet subspace onto l2(R). However, this operation is not an isometry of a general wavelet subspace onto l2(R). As a result, many sampling theories only concentrate on the recovery of a signal in a single wavelet subspace. In this paper, some theorems are proposed to discuss the isometric isomorphism of a wavelet subspace and a convolved l2(R) space. We show that the sampling operation is an isometric isomorphism of a wavelet subspace onto a convolved l2(R) space only if the sampling operation is an isomorphism of a wavelet subspace onto l2(R). Based on the isometric isomorphism, we further verify the existence of the mapping from the samples to the projection of a signal on an approximation space. At last, we propose the corresponding algorithm to construct this mapping so that the optimal approximations of a signal at the different resolution can be recovered from the samples. The simulation shows that our algorithm is more suitable to recover the projection of a signal than Shannon sampling theorem in a general multiresolution analysis.