An analogue of Bratteli-Jorgensen loop group actions for GMRA's

Details An analogue of Bratteli-Jorgensen loop group actions for GMRA's An analogue of Bratteli-Jorgensen loop group actions for GMRA's

2003-08-14

arxiv.org/abs/math/0308131v1

Wavelets, Frames, and Operator Theory (College Park, Maryland, January 15-21, 2003) (C. Heil, P.E.T. Jorgensen, and D. Larson,

Mathematics

Operator Algebras

15 pages; AMS-LaTeX; submitted to proceedings of AMS Special Session on Wavelets, Frames, and Operator Theory held at Baltimor

Scientific paper

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 L^2(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 L^2(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.

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