A necessary and sufficient condition for the existence of a father wavelet

Volume 114 / 1995

Gustaf Gripenberg Studia Mathematica 114 (1995), 207-226 DOI: 10.4064/sm-114-3-207-226

Abstract

It is proved that if ${2^{-m/2} ψ(2^{-m} • - k)}_{m,k ∈ ℤ}$ is an orthonormal basis in $L^2(ℝ;ℂ)$, then the mother wavelet ψ is obtained from a multiresolution generated by a father wavelet if and only if $∑_{p=1}^{∞| ∑_{k ∈ ℤ} |ψ̂(2^{p}(• + k))|^2 > 0$ a.e.

Authors

  • Gustaf Gripenberg

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