One of the major differences between Paley-Wiener spaces of bandlimited signals and the principal shift-invariant (PSI) spaces of wavelet theory is that the latter, although shift-invariant, are in general not translation-invariant. In this paper we study the extra difficulties non-translation-invariance creates for the sampling theory of PSI and multiresolution spaces. In particular it is shown that sampling in PSI spaces requires an extra initialization step to determine the times at which sampled data is acquired. An algorithm is developed to provide this initialization and its effectiveness shown theoretically and demonstrated on a synthetic data set.
Acta Applicandae Mathematicae Vol. 107, Issue 1-3, p. 373-398