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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 $m\sb 0(\omega)=\sum\sb n h(n)\exp(-ni\omega)$ satisfies the conditions $\vert m\sb 0(\omega)\vert\sp 2+\vert m\sb 0(\omega+\pi)\vert\sp 2=1$, $m\sb 0(0)=1$. A necessary and sufficient condition is given in order that the right frame of wavelets constructed from $h$ be not an orthonormal basis of $L\sp 2(R)$. The mapping from sequences to wavelets defines a continuous map from a subset of $\ell\sp 2(Z)$ into $L\sp 2(R)$.

46B15Summability and bases in normed spaces
42B10Fourier type transforms, several variables
42C40Wavelets and other special systems
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