Isometric isomorphism between a Hilbert space of sequence and a general wavelet subspace

The success of typical wavelet sampling theories mostly benefits from the isomorphism Tf={f(k)}_k between a wavelet subspace and l²(R), but, due to the ignorance of isometry, their main results only concentrate on the recovery of signal in a single wavelet subspace. Here, some theorems are proposed to discuss the isometric isomorphism of a wavelet subspace and a convolution weighted l²(R) space where Tf={f(k)}_k is a distance-preserving map. In the simulation, we show that the projection of signal on the subspace, instead of only signal itself, is recovered from the samples due to the isometric isomorphism between a wavelet subspace and a convolution weighted l²(R) space.