Irregular sampling theorems for wavelets subspaces

For finite energy 7-band continuous signal f ( t) , t E R, i.e., f E L 2 ( R) and suppf(w) = [-7,-y], the classical Shannon Sampling Theorem gave the following reconstruction formula, where f ( w) is the Fourier transform of f ( t) defined by f ( w) = J, f ( t)e-*”*dt. Unfortunately it is not appropriate for non-band-limited signals. However if we let y = 2 m ~ , m € 2, this problem can be viewed as a special case of sampling in wavelet subspaces with p(t) = sin?rt/nt playing the role of scaling function of MRA {Vm = W ( ~ ( 2 ~ t- n},},. Re-alizing these properties, Walter[G] extended (1) to a class of wavelet subspaces. Let cp(t) be a continuous scaling function of MRA(V,}, such that cp(t)l < O(ltl-’-“) for some E> 0 when It1-+ 00. Walter[G] showed that, in orthonormal case, there is a sequence {SR ( t)}k in VO such that S k ( t) = so(t- k) and f(t) = f ( k) S O ( t- k) for f E Vo. (2