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Orthonormal MRA wavelets: spectral formulas and algorithms. (English) Zbl 1245.42028

The authors present and develop an approach to orthonormal wavelets obtained from a multiresolution analysis based on the idea that functions in \(L^2(\mathbb R)\) can be given in terms of an orthonormal basis of \(L^2[0,1)\) and integer translations thereof as well as of an orthonormal basis of \(L^2[\pm 1,\pm 2)\) and dyadic dilations of these bases. Thus the relations between the scaling function and the wavelet and algorithms for computing these are expressed in terms of the coefficients in the representation between these two expansions. The case where the bases in \(L^2[0,1)\) and \(L^2[\pm 1,\pm 2)\) are taken to be the Haar bases are studied in more detail.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
47N40 Applications of operator theory in numerical analysis
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References:

[1] Birman M. S., Spectral Theory of Self-Adjoint Operators in Hilbert Space (1987) · Zbl 0744.47017
[2] Dai X., Mem. Amer. Math. Soc. 134 pp viii+68–
[3] DOI: 10.1002/cpa.3160410705 · Zbl 0644.42026 · doi:10.1002/cpa.3160410705
[4] DOI: 10.1137/1.9781611970104 · Zbl 0776.42018 · doi:10.1137/1.9781611970104
[5] DOI: 10.1137/0523059 · Zbl 0788.42013 · doi:10.1137/0523059
[6] DOI: 10.1007/s00041-010-9148-z · Zbl 1232.42033 · doi:10.1007/s00041-010-9148-z
[7] DOI: 10.1016/j.acha.2010.08.006 · Zbl 1211.42031 · doi:10.1016/j.acha.2010.08.006
[8] DOI: 10.1090/S0002-9947-1993-1117215-0 · doi:10.1090/S0002-9947-1993-1117215-0
[9] DOI: 10.1201/9781420049985 · doi:10.1201/9781420049985
[10] DOI: 10.1063/1.529093 · Zbl 0757.46012 · doi:10.1063/1.529093
[11] Mallat S., A Wavelet Tour of Signal Processing (1998) · Zbl 1125.94306
[12] Micchelli C. A., Linear Algebra Appl. 115 pp 841–
[13] DOI: 10.1090/conm/365/06706 · doi:10.1090/conm/365/06706
[14] DOI: 10.1142/S0219691306001385 · Zbl 1139.65340 · doi:10.1142/S0219691306001385
[15] DOI: 10.1006/jfan.1996.3079 · Zbl 0891.42018 · doi:10.1006/jfan.1996.3079
[16] Schipp F., Walsh Series. An Introduction to Dyadic Harmonic Analysis (1990)
[17] DOI: 10.1007/978-3-0348-8871-4_24 · doi:10.1007/978-3-0348-8871-4_24
[18] DOI: 10.1006/acha.1996.0015 · Zbl 0874.65104 · doi:10.1006/acha.1996.0015
[19] DOI: 10.2307/1969463 · Zbl 0034.06102 · doi:10.2307/1969463
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