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Necessary and sufficient conditions for constructing orthonormal wavelet bases. (English) Zbl 0757.46012
Let \(h\) be an absolutely summable complex sequence whose Fourier transform
\(m_ 0(\omega)=\sum_ n h(n)\exp(-ni\omega)\) satisfies the conditions \(| m_ 0(\omega)|^ 2+| m_ 0(\omega+\pi)|^ 2=1\), \(m_ 0(0)=1\). A necessary and sufficient condition is given in order that the right frame of wavelets constructed from \(h\) be not an orthonormal basis of \(L^ 2(R)\). The mapping from sequences to wavelets defines a continuous map from a subset of \(\ell^ 2(Z)\) into \(L^ 2(R)\).

MSC:
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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