Decomposition of Integral Self-Affine Multi-Tiles

In this paper, we propose a method to decompose an integral self-affine ${\mathbb Z}^n$-tiling set $K$ into measure disjoint pieces $K_j$ satisfying $K=\displaystyle\bigcup K_j$ in such a way that the collection of sets $K_j$ forms an integral self-affine collection associated with the matrix $B$ and this with a minimum number of pieces $K_j$. When used on a given measurable $\mathbb{Z}^n$-tiling set $K\subset\mathbb{R}^n$, this decomposition terminates after finitely many steps if and only if the set $K$ is an integral self-affine multi-tile. Furthermore, we show that the minimal decomposition we provide is unique.Comment: 15pages, 5figures, added references, typo correction

Spectral measures associated with the factorization of the Lebesgue measure on a set via convolution

Let $Q$ be a fundamental domain of some full-rank lattice in ${\Bbb R}^d$ and let $\mu$ and $\nu$ be two positive Borel measures on ${\Bbb R}^d$ such that the convolution $\mu\ast\nu$ is a multiple of $\chi_Q$. We consider the problem as to whether or not both measures must be spectral (i.e. each of their respective associated $L^2$ space admits an orthogonal basis of exponentials) and we show that this is the case when $Q = [0,1]^d$. This theorem yields a large class of examples of spectral measures which are either absolutely continuous, singularly continuous or purely discrete spectral measures. In addition, we propose a generalized Fuglede's conjecture for spectral measures on ${\Bbb R}^1$ and we show that it implies the classical Fuglede's conjecture on ${\Bbb R}^1$

Frames of multi-windowed exponentials on subsets of Rd{\mathbb R}^d

Given discrete subsets $\Lambda_j\subset {\Bbb R}^d$, $j=1,...,q$, consider the set of windowed exponentials $\bigcup_{j=1}^{q}\{g_j(x)e^{2\pi i }: \lambda\in\Lambda_j\}$ on $L^2(\Omega)$. We show that a necessary and sufficient condition for the windows $g_j$ to form a frame of windowed exponentials for $L^2(\Omega)$ with some $\Lambda_j$ is that $m\leq \max_{j\in J}|g_j|\leq M$ almost everywhere on $\Omega$ for some subset $J$ of $\{1,..., q\}$. If $\Omega$ is unbounded, we show that there is no frame of windowed exponentials if the Lebesgue measure of $\Omega$ is infinite. If $\Omega$ is unbounded but of finite measure, we give a sufficient condition for the existence of Fourier frames on $L^2(\Omega)$. At the same time, we also construct examples of unbounded sets with finite measure that have no tight exponential frame

Nonuniform wavelets and wavelet sets related to one-dimensional spectral pairs

AbstractA generalization of Mallat's classical multiresolution analysis, based on the theory of spectral pairs, was considered in two articles by Gabardo and Nashed. In this setting, the associated translation set is no longer a discrete subgroup of R but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. As a generalization of Dai, Larson, and Speegle's theory of wavelet sets, we prove in this paper the existence of nonuniform wavelet sets associated with the same translation and dilation parameters

Undecidable properties of self-affine sets and multi-tape automata

We study the decidability of the topological properties of some objects coming from fractal geometry. We prove that having empty interior is undecidable for the sets defined by two-dimensional graph-directed iterated function systems. These results are obtained by studying a particular class of self-affine sets associated with multi-tape automata. We first establish the undecidability of some language-theoretical properties of such automata, which then translate into undecidability results about their associated self-affine sets.Comment: 10 pages, v2 includes some corrections to match the published versio