eBook ISBN:  9781470419653 
Product Code:  MEMO/233/1097.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 
eBook ISBN:  9781470419653 
Product Code:  MEMO/233/1097.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 233; 2015; 85 ppMSC: Primary 42; Secondary 52
There exist results on the connection between the theory of wavelets and the theory of integral selfaffine tiles and in particular, on the construction of wavelet bases using integral selfaffine tiles. However, there are many nonintegral selfaffine tiles which can also yield wavelet basis. In this work, the author gives a complete characterization of all one and two dimensional \(A\)dilation scaling sets \(K\) such that \(K\) is a selfaffine tile satisfying \(BK=(K+d_1)\bigcup (K+d_2)\) for some \(d_1,d_2\in\mathbb{R}^2\), where \(A\) is a \(2\times 2\) integral expansive matrix with \(\lvert \det A\rvert=2\) and \(B=A^t\).

Table of Contents

Chapters

1. Introduction

2. Preliminary Results

3. A sufficient condition for a selfaffine tile to be an MRA scaling set

4. Characterization of the inclusion $K\subset BK$

5. Selfaffine scaling sets in $\mathbb {R}^2$: the case $0\in \mathcal {D}$

6. Selfaffine scaling sets in $\mathbb {R}^2$: the case $\mathcal {D}=\{d_1,d_2\}\subset \mathbb {R}^2$

7. Conclusion


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There exist results on the connection between the theory of wavelets and the theory of integral selfaffine tiles and in particular, on the construction of wavelet bases using integral selfaffine tiles. However, there are many nonintegral selfaffine tiles which can also yield wavelet basis. In this work, the author gives a complete characterization of all one and two dimensional \(A\)dilation scaling sets \(K\) such that \(K\) is a selfaffine tile satisfying \(BK=(K+d_1)\bigcup (K+d_2)\) for some \(d_1,d_2\in\mathbb{R}^2\), where \(A\) is a \(2\times 2\) integral expansive matrix with \(\lvert \det A\rvert=2\) and \(B=A^t\).

Chapters

1. Introduction

2. Preliminary Results

3. A sufficient condition for a selfaffine tile to be an MRA scaling set

4. Characterization of the inclusion $K\subset BK$

5. Selfaffine scaling sets in $\mathbb {R}^2$: the case $0\in \mathcal {D}$

6. Selfaffine scaling sets in $\mathbb {R}^2$: the case $\mathcal {D}=\{d_1,d_2\}\subset \mathbb {R}^2$

7. Conclusion