Two-channel sampling in wavelet subspaces

We develop two-channel sampling theory in the wavelet subspace V1 from the multi resolution analysis {Vj}j∈Z. Extending earlier results by G. G. Walter [11], W. Chen and S. Itoh [2] and Y. M. Hong et al [5] on the sampling theory in the wavelet or shift invariant spaces, we find a necessary and sufficient condition for two-channel sampling expansion formula to hold in V1. 1 Indroduction The classical Whittaker-Shannon-Kotel’nikov(WSK) sampling theorem [4] states that any signal f(t) with finite energy and the bandwidth π can be completely reconstructed from its discrete values by the formula f(t) = ∞ ∑ n=−∞ f(n) sinπ(t− n) π(t− n) . WSK sampling theorem has been extended in many directions (see [1], [2], [5], [6], [7], [8], [10], [11], [12] and references therein). G. G. Walter [11] developed a sampling theorem in wavelet subspaces, noting that the sampling function sinct := sinπt/πt in the WSK sampling theorem is a scaling function of a multi-resolution analysis. A. J. E. M. Janssen [6] used the Zak transform to generalize Walter’s work to regular shifted sampling. Later, W. Chen and S. Itoh [2] (see also [12]) extended Walter’s result further by relaxing conditions