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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 (ω)= n h(n)exp(-niω) satisfies the conditions |m 0 (ω)| 2 +|m 0 (ω+π)| 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 2 (Z) into L 2 (R).


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