An analogue of Bratteli-Jorgensen loop group actions for GMRA’s
Lawrence W. Baggett, Palle E. T. Jørgensen, Kathy D. Merrill, Judith A. Packer · Contemporary mathematics - American Mathematical Society · 2004
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.