Representation theory and multilevel filters

2022/08/30 by Alpay, Daniel, Jorgensen, Palle, Lewkowicz, Izchak
#46L10 #46L54 #60J25 #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.2208.13990

Abstract

We present a general setting where wavelet filters and multiresolution decompositions can be defined, beyond the classical \mathbf L²(\mathbb R,dx) setting. This is done in a framework of \em iterated function system (IFS) measures; these include all cases studied so far, and in particular the Julia set/measure cases. Every IFS has a fixed order, say N, and we show that the wavelet filters are indexed by the infinite dimensional group G of functions from X into the unitary group U_N. We call G the loop group because of the special case of the unit circle.

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