# On the existence of wavelets for non-expansive dilation matrices.

Collectanea Mathematica (2003)

• Volume: 54, Issue: 2, page 163-179
• ISSN: 0010-0757

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## Abstract

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Sets which simultaneously tile Rn by applying powers of an invertible matrix and translations by a lattice are studied. Diagonal matrices A for which there exist sets that tile by powers of A and by integer translations are characterized. A sufficient condition and a necessary condition on the dilations and translations for the existence of such sets are also given. These conditions depend in an essential way on the interplay between the eigenvectors of the dilation matrix and the translation lattice rather than the usual dependence on the eigenvalues. For example, it is shown that for any values |a| &gt; 1 &gt; |b|, there is a (2 x 2) matrix A with eigenvalues a and b for which such a set exists, and a matrix A' with eigenvalues a and b for which no such set exists. Finally, these results are related to the existence of wavelets for non-expansive dilations.

## How to cite

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Speegle, Darrin. "On the existence of wavelets for non-expansive dilation matrices.." Collectanea Mathematica 54.2 (2003): 163-179. <http://eudml.org/doc/43075>.

@article{Speegle2003,
abstract = {Sets which simultaneously tile Rn by applying powers of an invertible matrix and translations by a lattice are studied. Diagonal matrices A for which there exist sets that tile by powers of A and by integer translations are characterized. A sufficient condition and a necessary condition on the dilations and translations for the existence of such sets are also given. These conditions depend in an essential way on the interplay between the eigenvectors of the dilation matrix and the translation lattice rather than the usual dependence on the eigenvalues. For example, it is shown that for any values |a| &gt; 1 &gt; |b|, there is a (2 x 2) matrix A with eigenvalues a and b for which such a set exists, and a matrix A' with eigenvalues a and b for which no such set exists. Finally, these results are related to the existence of wavelets for non-expansive dilations.},
author = {Speegle, Darrin},
journal = {Collectanea Mathematica},
keywords = {Ondículas; Teselaciones; Matrices de dilatación; tiling; group actions; orthonormal wavelet; MSF wavelet; non-expansive dilations},
language = {eng},
number = {2},
pages = {163-179},
title = {On the existence of wavelets for non-expansive dilation matrices.},
url = {http://eudml.org/doc/43075},
volume = {54},
year = {2003},
}

TY - JOUR
AU - Speegle, Darrin
TI - On the existence of wavelets for non-expansive dilation matrices.
JO - Collectanea Mathematica
PY - 2003
VL - 54
IS - 2
SP - 163
EP - 179
AB - Sets which simultaneously tile Rn by applying powers of an invertible matrix and translations by a lattice are studied. Diagonal matrices A for which there exist sets that tile by powers of A and by integer translations are characterized. A sufficient condition and a necessary condition on the dilations and translations for the existence of such sets are also given. These conditions depend in an essential way on the interplay between the eigenvectors of the dilation matrix and the translation lattice rather than the usual dependence on the eigenvalues. For example, it is shown that for any values |a| &gt; 1 &gt; |b|, there is a (2 x 2) matrix A with eigenvalues a and b for which such a set exists, and a matrix A' with eigenvalues a and b for which no such set exists. Finally, these results are related to the existence of wavelets for non-expansive dilations.
LA - eng
KW - Ondículas; Teselaciones; Matrices de dilatación; tiling; group actions; orthonormal wavelet; MSF wavelet; non-expansive dilations
UR - http://eudml.org/doc/43075
ER -

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