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# memo_has_moved_text();Self-affine scaling sets in $\mathbb {R}^2$

Xiaoye Fu and Jean-Pierre Gabardo

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 233, Number 1097
ISBNs: 978-1-4704-1091-9 (print); 978-1-4704-1965-3 (online)
DOI: http://dx.doi.org/10.1090/memo/1097
Published electronically: May 19, 2014
Keywords:(MRA) scaling set, self-affine tile

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Chapters

• Chapter 1. Introduction
• Chapter 2. Preliminary Results
• Chapter 3. A sufficient condition for a self-affine tile to be an MRA scaling set
• Chapter 4. Characterization of the inclusion $K\subset BK$
• Chapter 5. Self-affine scaling sets in $\mathbb {R}^2$: the case $0\in \mathcal {D}$
• Chapter 6. Self-affine scaling sets in $\mathbb {R}^2$: the case $\mathcal {D}=\{d_1,d_2\}\subset \mathbb {R}^2$
• Chapter 7. Conclusion

### Abstract

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, we give 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 $B=A^t$.