Frames and sampling theorems for translation-invariant subspaces

A sampling theorem for frames in translation-invariant subspaces of L/sup 2/(R) is established, which is a generalization of a result of W. Chen and S. Itoh for Riesz bases (see IEEE Trans. Sig. Processing, vol.46, p.2822-4, 1998). Then, some necessary conditions for sampling are derived for Riesz bases. Some relationships and properties are also derived about the relevant functions.