Representation theory and multilevel filters

Daniel Alpay; Palle Jorgensen; Izchak Lewkowicz

doi:10.1007/s12190-022-01805-z

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Representation theory and multilevel filters

Journal article

Open access

Peer reviewed

Representation theory and multilevel filters

Daniel Alpay, Palle Jorgensen and Izchak Lewkowicz

Journal of applied mathematics & computing, Vol.69(2), pp.1599-1657

04/2023

DOI: 10.1007/s12190-022-01805-z

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Abstract

We present a general setting where wavelet filters and multiresolution decompositions can be defined, beyond the classical $\mathbf L^2(\mathbb R,dx)$ 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.

Computational Mathematics

Applied Mathematics

46L54, 46L10, 60J25

Functional Analysis (math.FA)

Probability (math.PR)

FOS: Mathematics

Mathematics - Probability

Mathematics - Functional Analysis

Details

Title: Subtitle: Representation theory and multilevel filters
Creators: Daniel Alpay
Palle Jorgensen
Izchak Lewkowicz
Resource Type: Journal article
Publication Details: Journal of applied mathematics & computing, Vol.69(2), pp.1599-1657
DOI: 10.1007/s12190-022-01805-z
ISSN: 1598-5865
eISSN: 1865-2085
Publisher: Springer Science and Business Media LLC
Language: English
Electronic publication date: 10/25/2022
Date published: 04/2023
Academic Unit: Mathematics
Record Identifier: 9984532059302771

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