eBook ISBN:  9781470403133 
Product Code:  MEMO/152/720.E 
List Price:  $49.00 
MAA Member Price:  $44.10 
AMS Member Price:  $29.40 
eBook ISBN:  9781470403133 
Product Code:  MEMO/152/720.E 
List Price:  $49.00 
MAA Member Price:  $44.10 
AMS Member Price:  $29.40 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 152; 2001; 60 ppMSC: Primary 46; 42; 43; Secondary 41
Let \(N\in\mathbb{N}\), \(N\geq2\), be given. Motivated by wavelet analysis, we consider a class of normal representations of the \(C^{\ast}\)algebra \(\mathfrak{A}_{N}\) on two unitary generators \(U\), \(V\) subject to the relation \(UVU^{1}=V^{N}\). The representations are in onetoone correspondence with solutions \(h\in L^{1}\left(\mathbb{T}\right)\), \(h\geq0\), to \(R\left(h\right)=h\) where \(R\) is a certain transfer operator (positivitypreserving) which was studied previously by D. Ruelle. The representations of \(\mathfrak{A}_{N}\) may also be viewed as representations of a certain (discrete) \(N\)adic \(ax+b\) group which was considered recently by J.B. Bost and A. Connes.
ReadershipGraduate students and research mathematicians interested in functional analysis.

Table of Contents

Chapters

1. Introduction

2. A discrete $ax + b$ group

3. Proof of Theorem 2.4

4. Wavelet filters

5. Cocycle equivalence of filter functions

6. The transfer operator of Keane

7. A representation theorem for $R$harmonic functions

8. Signed solutions to $R(f) = f$


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Let \(N\in\mathbb{N}\), \(N\geq2\), be given. Motivated by wavelet analysis, we consider a class of normal representations of the \(C^{\ast}\)algebra \(\mathfrak{A}_{N}\) on two unitary generators \(U\), \(V\) subject to the relation \(UVU^{1}=V^{N}\). The representations are in onetoone correspondence with solutions \(h\in L^{1}\left(\mathbb{T}\right)\), \(h\geq0\), to \(R\left(h\right)=h\) where \(R\) is a certain transfer operator (positivitypreserving) which was studied previously by D. Ruelle. The representations of \(\mathfrak{A}_{N}\) may also be viewed as representations of a certain (discrete) \(N\)adic \(ax+b\) group which was considered recently by J.B. Bost and A. Connes.
Graduate students and research mathematicians interested in functional analysis.

Chapters

1. Introduction

2. A discrete $ax + b$ group

3. Proof of Theorem 2.4

4. Wavelet filters

5. Cocycle equivalence of filter functions

6. The transfer operator of Keane

7. A representation theorem for $R$harmonic functions

8. Signed solutions to $R(f) = f$