Acta Appl Math (2009) 107: 373–398 DOI 10.1007/s10440-009-9480-y Non-Translation-Invariance and the Synchronization Problem in Wavelet Sampling Jeffrey Hogan · Joseph Lakey Received: 21 October 2008 / Accepted: 12 February 2009 / Published online: 5 March 2009 © Springer Science+Business Media B.V. 2009 Abstract 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 diffi- culties 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. Keywords Sampling · Principal shift-invariant spaces · Multiresolution analysis · Zak transform · Scaling functions · Quadrature mirror filter Mathematics Subject Classification (2000) Primary 94A20 · Secondary 42C40 1 Introduction The traditional model space of signal analysis is the Paley-Wiener space PW  of those L 2 -signals f which are bandlimited to the interval [−/2,/2] in the sense that their This paper was partly written while Jeff Hogan was on sabbatical at the University of Vienna, where he was supported by the European Union EUCETIFA grant MEXT-CT-2004-517154 and a University of Arkansas Fulbright College of Arts and Sciences Research Incentive Grant. J. Hogan ( ) School of Mathematical and Physical Sciences, University of Newcastle, NSW 2308, Newcastle, Australia e-mail: Jeff.Hogan@newcastle.edu.au J. Lakey Department of Mathematical Sciences, New Mexico State University, P.O. Box 30001 Department 3MB, Las Cruces, NM 88003-8001, USA e-mail: jlakey@nmsu.edu