Oversampling of wavelet frames for real dilations (Q2890311)

Oversampling of wavelet frames has been a subject of intensive study by several authors, starting with the celebrated Second Oversampling Theorem of \textit{C. K. Chui} and \textit{X. Shi} [Appl. Comput. Harmon. Anal. 1, No. 1, 29--49 (1993; Zbl 0788.42011); Proc. Am. Math. Soc. 121, No. 2, 511--517 (1994; Zbl 0802.41015)]. The goal of the article under review is to extend this theorem to arbitrary real dilations. The paper is organized as follows. Section 1 is an introduction. Section 2 studies the notions of approximate transversals and approximate duals, concepts introduced by the authors. In proving one of the results in this section, the authors adopted the notion of a constellation, borrowed from coding theory as in the work of \textit{A. R. Calderbank} and \textit{N. J. A. Sloane} [IEEE Trans. Inf. Theory 33, 177--195 (1987; Zbl 0638.94018)]. Section 3 presents a generalization of the Second Sampling Theorem to real dilations. In Section 4, the authors present results on oversampling for dual affine frames, and show a counterexample to a result claimed by \textit{Ch. K. Chui, W. Czaja, M. Maggioni} and \textit{G. Weiss} [J. Fourier Anal. Appl. 8, No. 2, 173--200 (2002; Zbl 1005.42020)]. Finally, in Section 5 the authors show results on the equivalence of tight frame preservation for dilation matrix oversampling of the dilation matrix and membership in Behera-Weber class wavelets; c.f. \textit{B. Behera} [Bull. Lond. Math. Soc. 36, No. 2, 221--230 (2004; Zbl 1046.42025)]; \textit{E. Weber} [J. Fourier Anal. Appl. 6, No. 5, 551--558 (2000; Zbl 0960.42015)].