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

L. W. Baggett; P. E. T. Jorgensen; K. D. Merrill; J. A. Packer

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

L. W. Baggett·P. E. T. Jorgensen·K. D. Merrill·J. A. Packer

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Abstract

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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BibTeX RIS

L. W. Baggett, P. E. T. Jorgensen, K. D. Merrill, J. A. Packer. 2003-08-14. An analogue of Bratteli-Jorgensen loop group actions for GMRA's. https://arxiv.org/abs/math/0308131

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