Necessary and sufficient conditions for constructing orthonormal wavelet bases. (English) Zbl 0757.46012

Lawton, Wayne M.

J. Math. Phys. 32, No. 1, 57-61 (1991).

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 \(L^2(R)\).

Reviewer: Julian Musielak (Poznań)

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

Keywords:

absolutely summable complex sequence; Fourier transform; right frame of wavelets; mapping from sequences to wavelets

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