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The canonical dual frame of a wavelet frame. (English) Zbl 1013.42023
It is well known that there exist wavelet frames $$\{2^{j/2}\psi(2^jx-k)\}_{j,k\in Z}$$ for which the canonical dual frame does not have wavelet structure. In this paper the authors find such a frame with the additional property that other duals, which have the wavelet structure, exist; in fact, infinitely many such duals exist. Similar constructions are possible for wavelet frames $$\{2^{j/2}\psi_l(2^jx-k)\}_{j,k\in Z, l=1,\dots,r}$$ generated by $$r$$ functions. For such a frame, equivalent conditions for the canonical dual frame to be generated by $$2^Jr$$ functions (for some $$J=0,1,2,\dots$$) are given.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
##### Keywords:
wavelet frame; dual frame; canonical dual
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##### References:
 [1] Chui, C.K; Shi, X.L, Inequalities of littlewood – paley type for frames and wavelets, SIAM J. math. anal., 24, 263-277, (1993) · Zbl 0766.41013 [2] Daubechies, I, The wavelet transform, time-frequency localization and signal analysis, IEEE trans. inform. theory, 36, 961-1005, (1990) · Zbl 0738.94004 [3] Daubechies, I, Ten lectures on wavelets, CBMS-NSF regional conference series in applied mathematics, 61, (1992), SIAM Philadelphia [4] I. Daubechies, and, B. Han, Pairs of dual wavelet frames from any two refinable functions, Constr. Appr, to appear. · Zbl 1055.42025 [5] Duffin, R.J; Schaeffer, A.C, A class of nonharmonic Fourier series, Trans. amer. math. soc., 72, 341-366, (1952) · Zbl 0049.32401 [6] Han, B, On dual wavelet tight frames, Appl. comput. harmon. anal., 4, 380-413, (1997) · Zbl 0880.42017 [7] Naimark, M.A, Normed rings, (1959), Noordhoff Groningen [8] Ron, A; Shen, Z.W, Affine systems in L2($$R$$d). II. dual systems, J. Fourier anal. appl., 3, 617-637, (1997)
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