| eBook ISBN: | 978-1-4704-1965-3 |
| Product Code: | MEMO/233/1097.E |
| List Price: | $71.00 |
| MAA Member Price: | $63.90 |
| AMS Member Price: | $42.60 |
| eBook ISBN: | 978-1-4704-1965-3 |
| Product Code: | MEMO/233/1097.E |
| List Price: | $71.00 |
| MAA Member Price: | $63.90 |
| AMS Member Price: | $42.60 |
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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 self-affine tiles and in particular, on the construction of wavelet bases using integral self-affine tiles. However, there are many non-integral self-affine 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 self-affine 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 \(B=A^t\).
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Table of Contents
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Chapters
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1. Introduction
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2. Preliminary Results
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3. A sufficient condition for a self-affine tile to be an MRA scaling set
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4. Characterization of the inclusion $K\subset BK$
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5. Self-affine scaling sets in $\mathbb {R}^2$: the case $0\in \mathcal {D}$
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6. Self-affine scaling sets in $\mathbb {R}^2$: the case $\mathcal {D}=\{d_1,d_2\}\subset \mathbb {R}^2$
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7. Conclusion
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There exist results on the connection between the theory of wavelets and the theory of integral self-affine tiles and in particular, on the construction of wavelet bases using integral self-affine tiles. However, there are many non-integral self-affine 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 self-affine 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 self-affine tile to be an MRA scaling set
-
4. Characterization of the inclusion $K\subset BK$
-
5. Self-affine scaling sets in $\mathbb {R}^2$: the case $0\in \mathcal {D}$
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6. Self-affine scaling sets in $\mathbb {R}^2$: the case $\mathcal {D}=\{d_1,d_2\}\subset \mathbb {R}^2$
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7. Conclusion
