Self-Affine Scaling Sets in R2

There are many non-integral self-affine tiles which can 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 d1)(K d2) for some d1,d2?R2, where A is a 2×2 integral expansive matrix with detA=2 and B=At

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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 d1)⋃(K d2) for some d1,d2∈R2 , where A is a 2×2 integral expansive matrix with ∣detA∣=2 and B=At