2003/08/14 by L. W. Baggett, P. E. T. Jorgensen, K. D. Merrill +1
Mathematics · #math.OA #math.CA #msc:54C40 #msc:14E20 #msc:46E25 #msc:20C20
published as Wavelets, Frames, and Operator Theory (College Park, Maryland, January 15-21, 2003) (C. Heil, P.E.T. Jorgensen, and D. Larson, eds.), Contemp. Math., vol. 345, American Mathematical Society, Providence, RI, 2004, pp. 11-25 · 15 pages; AMS-LaTeX; submitted to proceedings of AMS Special Session on Wavelets, Frames, and Operator Theory held at Baltimore
arxiv created 2003/08/14 · arxiv updated 2009/12/01
Several years ago, O. Bratelli and P. Jorgensen developed the concept of m-systems of filters for dilation by a positive integer N>1 on L2(R). They constructed a loop group action on m-systems. By work of Mallat and Meyer, these m-systems are important in constructing multi-resolution analyses and wavelets associated to dilation by N and translation by Z on L2(R). In this paper, we discuss an extension of this loop-group construction to generalized filter systems, which we will call ``M-systems,'' associated with generalized multiresolution analyses. In particular, we show that every multiplicity function has an associated generalized loop group which acts freely and transitively on the set of M-systems corresponding to the multiplicity function. The results of Bratteli and Jorgensen correspond to the case where the multiplicity function is identically equal to 1.