Some applications of self-affine sets to wavelet theory

Abstract

<p>In this thesis, we study several applications of self-affine sets to wavelet theory. Five major topics are considered here: wavelet sets (scaling sets), multiwavelet sets (generalized scaling sets), self-affine tiles, integral self-affine multi-tiles, self-affine sets. We divide the thesis into six chapters to discuss these topics. In Chapter 1, the dimension function of a self-affine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK=(K+d_1)\bigcup(K+d_2)$, where $B=A^t$ and $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$. We show that it must be a constant in dimension $n=1$ or $2$ and it is bounded by $2\lvert K\rvert$ for any $n$. This result shows that all $A$-dilation self-affine scaling sets must be $A$-dilation MRA scaling sets in dimensions one and two. 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 Chapter 2 and Chapter 3, we give a complete characterization of all 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 an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$. In Chapter 2, we deal with a particular case where $0\in\{d_1,d_2\}$, i.e. a self-affine tile $K$ satisfies $BK=K\bigcup (K+d)$ for some $d\in\mathbb{R}^2$. Chapter 3 is devoted to the general case with $d_1, d_2\in\mathbb{R}^2$. Moreover, we give a sufficient condition for a self-affine tile, possibly non-integral, to be an MRA scaling set in Chapter 3. Gabardo and Yu first considered using integral self-affine tiles in the Fourier domain to construct wavelet sets and they produced a class of compact wavelet sets with certain self-similarity properties. In Chapter 4, we generalize their results to the integral self-affine multi-tiles setting. We characterize some analytic properties of integral self-affine multi-tiles under certain conditions. We also consider the problem of constructing (multi)wavelet sets using integral self-affine multi-tiles. Suppose that a measurable $\mathbb{Z}^n$-tiling set $K\subset\mathbb{R}^n$ is an integral self-affine multi-tile associated with an $n\times n$ integral expansive matrix $B$. To our knowledge, no one considered how to represent an integral self-affine $\mathbb{Z}^n$-tiling set as the disjoint union of prototiles. In Chapter 5, we provide an algorithm to decompose $K$ into disjoint pieces $K_j$ which satisfy $K=\displaystyle\bigcup K_j$ such that the collection of the sets $K_j$ is an integral self-affine collection associated with matrix $B$ and the number of pieces $K_j$ is minimal. Using this algorithm, we can determine whether a given measurable $\mathbb{Z}^n$-tiling set $K\subset\mathbb{R}^n$ is an integral self-affine multi-tile associated with any given $n\times n$ integral expansive matrix $B$. Furthermore, the minimal decomposition we provide is unique. Let $B$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of $\mathbb{R}^n$. The self-affine set $K=K(B,\mathcal{D})$ is the unique compact set satisfying the set equation $BK=\displaystyle\bigcup_{d\in\mathcal{D}}(K+d)$. In Chapter 6, we not only consider the problem how to compute the Lebesgue measure of self-affine sets $K(B,\mathcal{D})$, but also consider the Hausdorff measure for those with zero Lebesgue measure under the assumption that $K(B,\mathcal{D})$ is a self-similar set. In the case where $\text{card}(\mathcal{D})=\lvert\det B\rvert,$ we relate the Lebesgue measure of $K(B,\mathcal{D})$ to the upper Beurling density of the associated measure $\mu=\lim\limits_{s\to\infty}\sum\limits_{\ell_0,\dotsc,\ell_{s-1}\in\mathcal{D}}\delta_{\ell_0+B\ell_1+\dotsb+B^{s-1}\ell_{s-1}}.$ If, on the other hand, $\text{card}(\mathcal{D})<\lvert\det B\rvert$ and $B$ is a similarity matrix, we relate the Hausdorff measure $\mathcal{H}^s(K)$, where $s$ is the similarity dimension of $K$, to a corresponding notion of upper density for the measure $\mu$.</p>