Frequency-Scale Frames and the Solution of the Mexican Hat Problem

H. Q. Bui, R. S. Laugesen

Research output: Contribution to journalArticle

Abstract

We resolve a long-standing question on Lp completeness of the time-scale (or wavelet) system generated by the Mexican hat function, when p≥2. Our main result concerns frequency-scale systems generated by modulation and dilation of a single function. The mixed frame operator (analysis followed by synthesis) is shown to be bijective from Lq(ℝd) to itself, for 1≤q<∞, so that the frequency-scale synthesis operator is surjective. Tools include the discrete Calderón condition and a generalization of the Daubechies frame criterion in L2. Completeness of the Mexican hat and other wavelet systems in Lp then follows for 2≤p<∞, by Fourier imbedding of frequency-scale systems.

Original languageEnglish (US)
Pages (from-to)163-189
Number of pages27
JournalConstructive Approximation
Volume33
Issue number2
DOIs
StatePublished - Apr 1 2011

Keywords

  • Analysis
  • Completeness
  • Fourier space
  • Spanning
  • Synthesis
  • Wavelet

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Computational Mathematics

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