Journal of Fourier Analysis and Applications

, Volume 6, Issue 2, pp 153–170

Orthogonality criteria for compactly supported refinable functions and refinable function vectors

  • Jeffrey C. Lagarias
  • Yang Wang


A refinable function φ(x):ℝn→ℝ or, more generally, a refinable function vector Φ(x)=[φ1(x),...,φr(x)]T is an L1 solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding integer matrix. A refinable function vector is called orthogonal if {φj(x−α):α∈ℤn, 1≤j≤r form an orthogonal set of functions in L2(ℝn). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and multiwavelet bases of L2(ℝn). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported refinable functions and refinable function vectors.

Math Subject Classifications

42C10 42C15 

Keywords and Phrases

orthogonal refinable function orthogonal refinable function vector orthogonality criteria wavelet multiwavelet 


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Copyright information

© Birkhäuser 2000

Authors and Affiliations

  • Jeffrey C. Lagarias
    • 1
    • 2
  • Yang Wang
    • 1
    • 2
  1. 1.Information Science ResearchAT&T Labs-ResearchFlorham Park
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlanta

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