TY - JOUR
T1 - Completeness of Orthonormal Wavelet Systems for Arbitrary Real Dilations
AU - Laugesen, Richard S.
N1 - Funding Information:
My thanks go to Guido Weiss for encouraging this research, to Marcin Bownik for helpful feedback, and to the National Science Foundation for support under DMS-9970228.
PY - 2001/11
Y1 - 2001/11
N2 - 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.
AB - 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.
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U2 - 10.1006/acha.2001.0365
DO - 10.1006/acha.2001.0365
M3 - Article
AN - SCOPUS:0035508516
SN - 1063-5203
VL - 11
SP - 455
EP - 473
JO - Applied and Computational Harmonic Analysis
JF - Applied and Computational Harmonic Analysis
IS - 3
ER -