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A characterization of dimension functions of a class of semi‐orthogonal Parseval frame wavelets

Dimension functions play a significant role in the study of wavelets and have been attracting many waveletters’ interest. In recent years, the study of wavelet dimension functions has seen great achievements, but the study of Parseval frame wavelet (PFW) dimension functions has not. Bownik, Rzeszotnik and Speegle in 2001 and Arambašić, Bakić and Rajić in 2007 characterized Zd ‐periodic functions that are wavelet dimension functions. But it is open what a Zd ‐periodic function is qualified to be a dimension function of some semi‐orthogonal PFW. This paper addresses semi‐orthogonal PFW dimension functions associated with expansive matrices of determinant ± 2. We obtain a description of the ranges of semi‐orthogonal PFW dimension functions and establish a necessary and sufficient condition for an integer‐valued function to be a dimension function of some semi‐orthogonal PFW. Copyright © 2014 John Wiley & Sons, Ltd.