Diffraction of weighted lattice subsets

Weighted Dirac combs with pure point diffraction

A class of translation bounded complex measures, which have the form of weighted Dirac combs, on locally compact Abelian groups is investigated. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges as the Fourier transform of the autocorrelation measure. We present a sufficient set of conditions to ensure that the diffraction measure is a pure point measure. Simultaneously, we establish a natural link to the theory of the cut and project formalism and to the theory of almost periodic measures. Our conditions are general enough to cover the known theory of model …

Random point sets and their diffraction

The diffraction of various random subsets of the integer lattice $\mathbb{Z}^{d}$, such as the coin tossing and related systems, are well understood. Here, we go one important step beyond and consider random point sets in $\mathbb{R}^{d}$. We present several systems with an effective stochastic interaction that still allow for explicit calculations of the autocorrelation and the diffraction measure. We concentrate on one-dimensional examples for illustrative purposes, and briefly indicate possible generalisations to higher dimensions. In particular, we discuss the stationary Poisson process in…

Traces and extensions of certain weighted Sobolev spaces on $\mathbb{R}^n$ and Besov functions on Ahlfors regular compact subsets of $\mathbb{R}^n$

The focus of this paper is on Ahlfors $Q$-regular compact sets $E\subset\mathbb{R}^n$ such that, for each $Q-2<α\le 0$, the weighted measure $μ_α$ given by integrating the density $ω(x)=\text{dist}(x, E)^α$ yields a Muckenhoupt $\mathcal{A}_p$-weight in a ball $B$ containing $E$. For such sets $E$ we show the existence of a bounded linear trace operator acting from $W^{1,p}(B,μ_α)$ to $B^θ_{p,p}(E, \mathcal{H}^Q\vert_E)$ when $0<θ<1-\tfrac{α+n-Q}{p}$, and the existence of a bounded linear extension operator from $B^θ_{p,p}(E, \mathcal{H}^Q\vert_E)$ to $W^{1,p}(B, μ_α)$ when $1-\tfrac{α+n-Q}{p}…

超越树：加权汉明立方体子集的度量几何

Beyond trees: the metric geometry of subsets of weighted Hamming cubes

本文研究加权汉明立方体子集的度量几何，揭示其关联几何量间的意外关系，推导距离矩阵行列式等公式，拓展此前的树与标准汉明立方体相关结果，并刻画其中的非平凡1-多边形等式。