Construction of Scaling Partitions of Unity

Ole Christensen, Say Song Goh

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Partitions of unity in ℝd formed by (matrix) scales of a fixed function appear in many parts of harmonic analysis, e.g., wavelet analysis and the analysis of Triebel-Lizorkin spaces. We give a simple characterization of the functions and matrices yielding such a partition of unity. For expanding matrices, the characterization leads to easy ways of constructing appropriate functions with attractive properties like high regularity and small support. We also discuss a class of integral transforms that map functions having the partition of unity property to functions with the same property. The one-dimensional version of the transform allows a direct definition of a class of nonuniform splines with properties that are parallel to those of the classical B-splines. The results are illustrated with the construction of dual pairs of wavelet frames.
Original languageEnglish
Article number21
JournalFrontiers in Applied Mathematics and Statistics
Number of pages16
Publication statusPublished - 2017


  • Partition of unity
  • Splines
  • Wavelet frames
  • Dual frames
  • Integral transforms

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