Characterization of tile digit sets with prime determinants
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AbstractFor an expanding integral s x s matrix A with \det A\ = p, it is well known that if D = {d(0),...,d(p-1)} subset of Z(s) is a complete set of coset representatives of Z(s)/AZ(s), then T(A, D) is a self-affine tile. In this paper we show that if p is a prime, such D actually characterizes the tile digit sets provided that span(D) = R-s. This result is known for s = 1. the one-dimensional case [R. Kenyon, in: Contemp. Math., vol. 135, 1992, pp. 239-264] and the question for s > 1 has been considered by Lagarias and Wang [J. London Math. Soc. 53 (1996) 21-49] under some other conditions. The proof here involves a new setup to study the zeros of the mask m(xi) = p(-1) Sigma(j=0)(p-1) e(2pii(xi,d j)). It can also be generalized to consider the existence of a compactly supported L-1 -solution of the refinement equation (scaling function) with positive coefficients. (C) 2004 Published by Elsevier Inc.
All Author(s) ListHe XG, Lau KS
Journal nameApplied and Computational Harmonic Analysis
Volume Number16
Issue Number3
Pages159 - 173
LanguagesEnglish-United Kingdom
Keywordsdigit set; prime; refinement equation; root of unity; scaling function; self-affine tile; tile digit set
Web of Science Subject CategoriesMathematics; Mathematics, Applied; MATHEMATICS, APPLIED; Physics; Physics, Mathematical; PHYSICS, MATHEMATICAL

Last updated on 2021-22-01 at 00:31