Journal of Fourier Analysis and Applications

, Volume 1, Issue 2, pp 131–170

Self-Similar Lattice Tilings

Authors

  • Karlheinz Grochenig
    • Department of Mathematics U-9, University of Connecticut, Storrs, Connecticut 06269-3009
  • Andrew Haas
    • Department of Mathematics U-9, University of Connecticut, Storrs, Connecticut 06269-3009
Article

DOI: 10.1007/s00041-001-4007-6

Cite this article as:
Grochenig, K. & Haas, A. J Fourier Anal Appl (1994) 1: 131. doi:10.1007/s00041-001-4007-6

Abstract

We study the general question of the existence of self-similar lattice tilings of Euclidean space. A necessary and sufficient geometric condition on the growth of the boundary of approximate tiles is reduced to a problem in Fourier analysis that is shown to have an elegant simple solution in dimension one. In dimension two we further prove the existence of connected self-similar lattice tilings for parabolic and elliptic dilations. These results apply to produce Haar wavelet bases and certain canonical number systems.

Copyright information

© Birkhäuser Boston 1994