TY - JOUR
T1 - Self-similar lattice tilings and subdivision schemes
AU - Zhou, Ding-Xuan
PY - 2001
Y1 - 2001
N2 - Let M ∈ ℤs × s be a dilation matrix and let D ⊂ ℤs be a complete set of representatives of distinct cosets of ℤs/Mℤs. The self-similar tiling associated with M and D is the subset of Rs given by T(M, D) = {∑j=1
∞ M-j αj : αj ∈ D}. The purpose of this paper is to characterize self-similar lattice tilings, i.e., tilings T(M, D) which have Lebesgue measure one. In particular, it is shown that T(M, D) is a lattice tiling if and only if there is no nonempty finite set Λ ⊂ ℤs / (D - D) such that M-1 ((D - D) + Λ) ∩ ℤs ⊂ Λ. This set A can be restricted to be contained in a finite set K depending only on M and D. We also give a new proof for the fact that T(M, D) is a lattice tiling if and only if ∪n=1
∞ (∑j=0
n=1 Mj (D - D)) = Zs. Two approaches are provided, one based on scrambling matrices and the other based on primitive matrices. These will follow from the characterization of subdivision schemes associated with nonnegative masks in terms of finite powers of finite matrices, without computing eigenvalues or spectral radii. Our characterization shows that the convergence of the Subdivision scheme with a nonnegative mask depends only on the location of its positive coefficients.
AB - Let M ∈ ℤs × s be a dilation matrix and let D ⊂ ℤs be a complete set of representatives of distinct cosets of ℤs/Mℤs. The self-similar tiling associated with M and D is the subset of Rs given by T(M, D) = {∑j=1
∞ M-j αj : αj ∈ D}. The purpose of this paper is to characterize self-similar lattice tilings, i.e., tilings T(M, D) which have Lebesgue measure one. In particular, it is shown that T(M, D) is a lattice tiling if and only if there is no nonempty finite set Λ ⊂ ℤs / (D - D) such that M-1 ((D - D) + Λ) ∩ ℤs ⊂ Λ. This set A can be restricted to be contained in a finite set K depending only on M and D. We also give a new proof for the fact that T(M, D) is a lattice tiling if and only if ∪n=1
∞ (∑j=0
n=1 Mj (D - D)) = Zs. Two approaches are provided, one based on scrambling matrices and the other based on primitive matrices. These will follow from the characterization of subdivision schemes associated with nonnegative masks in terms of finite powers of finite matrices, without computing eigenvalues or spectral radii. Our characterization shows that the convergence of the Subdivision scheme with a nonnegative mask depends only on the location of its positive coefficients.
KW - Column-stochastic matrices
KW - Primitive matrices
KW - Scrambling matrices
KW - Self-similar lattice tilings
KW - Subdivision schemes
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U2 - 10.1137/S0036141000367977
DO - 10.1137/S0036141000367977
M3 - 21_Publication in refereed journal
VL - 33
SP - 1
EP - 15
JO - SIAM Journal on Mathematical Analysis
JF - SIAM Journal on Mathematical Analysis
SN - 0036-1410
IS - 1
ER -