Self-Affine Scaling Sets in $\mathbb {R}^2$
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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 $B=A^t$.
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