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Self-Affine Scaling Sets in $\mathbb{R}^2$
 
Xiaoye Fu The Chinese University of Hong Kong, Shatin, Hong Kong
Jean-Pierre Gabardo McMaster University, Hamilton, ON, Canada
Self-Affine Scaling Sets in R^2
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
Self-Affine Scaling Sets in R^2
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Self-Affine Scaling Sets in $\mathbb{R}^2$
Xiaoye Fu The Chinese University of Hong Kong, Shatin, Hong Kong
Jean-Pierre Gabardo McMaster University, Hamilton, ON, Canada
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
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2332015; 85 pp
    MSC: 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 \(B=A^t\).

  • Table of Contents
     
     
    • 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 $0\in \mathcal {D}$
    • 6. Self-affine scaling sets in $\mathbb {R}^2$: the case $\mathcal {D}=\{d_1,d_2\}\subset \mathbb {R}^2$
    • 7. Conclusion
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2332015; 85 pp
MSC: 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 \(B=A^t\).

  • 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 $0\in \mathcal {D}$
  • 6. Self-affine scaling sets in $\mathbb {R}^2$: the case $\mathcal {D}=\{d_1,d_2\}\subset \mathbb {R}^2$
  • 7. Conclusion
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.