Completeness of Orthonormal Wavelet Systems for Arbitrary Real Dilations

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It is shown that the discrete Calderón condition characterizes completeness of orthonormal wavelet systems, for arbitrary real dilations. That is, if a>1,b>0, and the system Ψ={aj/2ψ(ajx-bk):j,k∈Z} is orthonormal in L2(R), then Ψ is a basis for L2(R) if and only if ∑j∈Zψ̂(ajξ)2=b for almost every ξ∈R. A new proof of the Second Oversampling Theorem is found, by similar methods.

Original languageEnglish (US)
Pages (from-to)455-473
Number of pages19
JournalApplied and Computational Harmonic Analysis
Issue number3
StatePublished - Nov 2001

ASJC Scopus subject areas

  • Applied Mathematics


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