**Memoirs of the American Mathematical Society**

2015;
85 pp;
Softcover

MSC: Primary 42;
Secondary 52

Print ISBN: 978-1-4704-1091-9

Product Code: MEMO/233/1097

List Price: $71.00

AMS Member Price: $42.60

MAA Member Price: $63.90

**Electronic ISBN: 978-1-4704-1965-3
Product Code: MEMO/233/1097.E**

List Price: $71.00

AMS Member Price: $42.60

MAA Member Price: $63.90

# Self-Affine Scaling Sets in \(\mathbb{R}^{2}\)

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*Xiaoye Fu; Jean-Pierre Gabardo*

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

# Table of Contents

## Self-Affine Scaling Sets in $\mathbb{R}^{2}$

- Cover Cover11 free
- Title page i2 free
- Chapter 1. Introduction 18 free
- Chapter 2. Preliminary Results 916
- Chapter 3. A sufficient condition for a self-affine tile to be an MRA scaling set 1522
- Chapter 4. Characterization of the inclusion 𝐾⊂𝐵𝐾 1926
- Chapter 5. Self-affine scaling sets in ℝ²: the case 0∈𝒟 2936
- Chapter 6. Self-affine scaling sets in ℝ²: the case 𝒟={𝒹₁,𝒹₂}⊂ℝ² 5360
- Chapter 7. Conclusion 8188
- Bibliography 8390
- Index 8592 free
- Back Cover Back Cover197