×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1137/0515056 · Zbl 0578.42007 · doi:10.1137/0515056
[2] Haar A., Math. Ann. 69 pp 336– (1989)
[3] DOI: 10.1016/0041-5553(62)90031-9 · Zbl 0163.39303 · doi:10.1016/0041-5553(62)90031-9
[4] DOI: 10.1016/0041-5553(64)90253-8 · Zbl 0148.39501 · doi:10.1016/0041-5553(64)90253-8
[5] DOI: 10.1090/S0025-5718-1977-0431719-X · doi:10.1090/S0025-5718-1977-0431719-X
[6] Calderón A. P., Studia Math. 24 pp 113– (1964)
[7] DOI: 10.1063/1.1664570 · Zbl 0162.58403 · doi:10.1063/1.1664570
[8] DOI: 10.1063/1.1664833 · Zbl 0184.54601 · doi:10.1063/1.1664833
[9] DOI: 10.1016/0370-1573(74)90023-4 · doi:10.1016/0370-1573(74)90023-4
[10] Burt P., IEEE Commun. Commun. 31 pp 482– (1983)
[11] DOI: 10.1145/245.247 · doi:10.1145/245.247
[12] DOI: 10.1137/0515056 · Zbl 0578.42007 · doi:10.1137/0515056
[13] DOI: 10.1063/1.526761 · Zbl 0571.22021 · doi:10.1063/1.526761
[14] Grossman A., Ann Inst. H. Poincaré 45 pp 293– (1986)
[15] DOI: 10.1063/1.527388 · Zbl 0608.46014 · doi:10.1063/1.527388
[16] DOI: 10.1109/18.57199 · Zbl 0738.94004 · doi:10.1109/18.57199
[17] DOI: 10.1063/1.526072 · doi:10.1063/1.526072
[18] Tchamitchian P., C. R. Acad. Sci. Paris 303 pp 215– (1986)
[19] DOI: 10.1016/0375-9601(89)90003-0 · doi:10.1016/0375-9601(89)90003-0
[20] DOI: 10.4171/RMI/22 · doi:10.4171/RMI/22
[21] Lemarié P. G., J. Math. Pures Appl. 67 pp 227– (1988)
[22] Jaffard S., C. R. Acad. Sci. Paris 308 pp 79– (1989)
[23] Jaffard S., J. Math. Pures Appl. 68 pp 95– (1989)
[24] DOI: 10.1007/BF01205550 · doi:10.1007/BF01205550
[25] DOI: 10.1007/BF01206144 · doi:10.1007/BF01206144
[26] DOI: 10.1109/34.192463 · Zbl 0709.94650 · doi:10.1109/34.192463
[27] DOI: 10.1109/29.45554 · doi:10.1109/29.45554
[28] DOI: 10.1002/cpa.3160410705 · Zbl 0644.42026 · doi:10.1002/cpa.3160410705
[29] DOI: 10.1109/TASSP.1986.1164832 · doi:10.1109/TASSP.1986.1164832
[30] DOI: 10.1109/29.32282 · doi:10.1109/29.32282
[31] DOI: 10.1109/29.32283 · doi:10.1109/29.32283
[32] Pollen D., J. Am. Math. Soc. 3 pp 611– (1990)
[33] DOI: 10.1063/1.528688 · Zbl 0708.46020 · doi:10.1063/1.528688
[34] Oseledec V. I., Trans. Moscow Math. Soc. 19 pp 197– (1968)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.