Lattice tiling and the Weyl—Heisenberg frames

2001 ◽  
Vol 11 (4) ◽  
pp. 742-758 ◽  
Author(s):  
D. Han ◽  
Y. Wang
Keyword(s):  
Lattice Tiling

scholarly journals No lattice tiling of Zn by Lee sphere of radius 2

2020 ◽  
Vol 171 ◽  
pp. 105157
Author(s):  
Ka Hin Leung ◽  
Yue Zhou
Keyword(s):  
Lattice Tiling

scholarly journals Lattice Tilings by Cubes: Whole, Notched and Extended

10.37236/1352 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Mihail Kolountzakis
Keyword(s):  
Fourier Transform ◽  
Harmonic Analysis ◽  
Equivalent Form ◽  
Unit Cube ◽  
Indicator Function ◽  
Geometric Method ◽  
Zero Set ◽  
New Class ◽  
Lattice Tiling ◽  
The Fourier Transform

We discuss some problems of lattice tiling via Harmonic Analysis methods. We consider lattice tilings of ${\bf R}^d$ by the unit cube in relation to the Minkowski Conjecture (now a theorem of Hajós) and give a new equivalent form of Hajós's theorem. We also consider "notched cubes" (a cube from which a rectangle has been removed from one of the corners) and show that they admit lattice tilings. This has also been been proved by S. Stein by a direct geometric method. Finally, we exhibit a new class of simple shapes that admit lattice tilings, the "extended cubes", which are unions of two axis-aligned rectangles that share a vertex and have intersection of odd codimension. In our approach we consider the Fourier Transform of the indicator function of the tile and try to exhibit a lattice of appropriate volume in its zero-set.

scholarly journals Lattice tiling and density conditions for subspace Gabor frames

2013 ◽  
Vol 265 (7) ◽  
pp. 1170-1189 ◽  
Author(s):  
Jean-Pierre Gabardo ◽  
Deguang Han ◽  
Yun-Zhang Li
Keyword(s):  
Gabor Frames ◽  
Lattice Tiling

scholarly journals Multiple Lattice Tilings in Euclidean Spaces

2018 ◽  
Vol 62 (4) ◽  
pp. 923-929
Author(s):  
Qi Yang ◽  
Chuanming Zong
Keyword(s):  
Euclidean Space ◽  
Convex Domain ◽  
Euclidean Plane ◽  
Dimensional Euclidean Space ◽  
Euclidean Spaces ◽  
Convex Domains ◽  
Lattice Tiling ◽  
Centrally Symmetric

AbstractIn 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results. Except for parallelograms and centrally symmetric hexagons, there are no other convex domains that can form two-, three- or four-fold lattice tilings in the Euclidean plane. However, there are both octagons and decagons that can form five-fold lattice tilings. Whenever $n\geqslant 3$, there are non-parallelohedral polytopes that can form five-fold lattice tilings in the $n$-dimensional Euclidean space.

Sign in / Sign up

Export Citation Format

Share Document