eBook ISBN: | 978-1-4704-0313-3 |
Product Code: | MEMO/152/720.E |
List Price: | $49.00 |
MAA Member Price: | $44.10 |
AMS Member Price: | $29.40 |
eBook ISBN: | 978-1-4704-0313-3 |
Product Code: | MEMO/152/720.E |
List Price: | $49.00 |
MAA Member Price: | $44.10 |
AMS Member Price: | $29.40 |
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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 one-to-one 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 (positivity-preserving) 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.
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Table of Contents
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Chapters
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1. Introduction
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2. A discrete $ax + b$ group
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3. Proof of Theorem 2.4
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4. Wavelet filters
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5. Cocycle equivalence of filter functions
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6. The transfer operator of Keane
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7. A representation theorem for $R$-harmonic functions
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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 one-to-one 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 (positivity-preserving) 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 $R(f) = f$