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On dual wavelet tight frames. (English) Zbl 0880.42017
Summary: A characterization of multivariate dual wavelet tight frames for any general dilation matrix is presented in this paper. As an application, Lawton’s result on wavelet tight frames in \(L^2(\mathbb{R})\) is generalized to the \(n\)-dimensional case. Two ways of constructing certain dual wavelet tight frames in \(L^2(\mathbb{R}^n)\) are suggested. Finally, examples of smooth wavelet tight frames in \(L^2(\mathbb{R})\) and \(H^2(\mathbb{R})\) are provided. In particular, an example is given to demonstrate that there is a function \(\psi\) whose Fourier transform is positive, compactly supported, and infinitely differentiable which generates a non-MRA wavelet tight frame in \(H^2(\mathbb{R})\).

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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[1] Auscher, P., Il n’existe pas de bases d’ondelettes regulières dans l’espace de HardyH^{2}R, C. R. Acad. Sci. Paris Ser. I, 315, 769-772, (1992) · Zbl 0758.42019
[2] P. Auscher, G. Weiss, M. V. Wickerhauser, 1992, Local sine and cosine bases of Coifman and Meyer and the construction of smooth wavelets, Wavelet: A Tutorial in Theory and Applications, 237, 578, Academic Press, San Diego · Zbl 0767.42009
[3] Bonami, A.; Soria, F.; Weiss, G., Band-limited wavelets, J. Geom. Anal., 3, 543-578, (1993) · Zbl 0811.42012
[4] Chui, C. K.; Shi, X., On a littlewood – paley identity and characterization of wavelets, Math. Anal. Appl., 177, 608-626, (1993) · Zbl 0782.42025
[5] Chui, C. K.; Shi, X., Bessel sequences and affine frames, Appl. Comput. Harmonic Anal., 1, 29-49, (1993) · Zbl 0788.42011
[6] Cohen, A.; Daubechies, I., A stability criterion for biorthogonal wavelet bases and their related subband coding scheme, Duke Math. J., 68, 313-335, (1992) · Zbl 0784.42022
[7] Cohen, A.; Daubechies, I.; Feauveau, J. C., Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 45, 485-560, (1992) · Zbl 0776.42020
[8] Daubechies, I., Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41, 906-996, (1988) · Zbl 0644.42026
[9] Daubechies, I., Ten Lectures on Wavelets, SIAM-NSF Regional Conference Series, 61, (1992), SIAM Philadelphia · Zbl 0776.42018
[10] Han, B., Some applications of projection operators in wavelets, Acta Math. Sinica, 11, 105-112, (1995) · Zbl 0831.42022
[11] Hernández, E.; Wang, X.; Weiss, G., Smoothing minimally supported frequency (MSF) wavelets, J. Fourier Anal. & Appl., 2, 329-340, (1996) · Zbl 0944.42021
[12] Jia, R. Q.; Micchelli, C. A., Using the refinement equation for the construction of prewavelets. V. extensibility of trigonometric polynomial, Computing, 48, 61-72, (1992) · Zbl 0765.65023
[13] Jia, R. Q.; Shen, Z. W., Multiresolution and wavelets, Proc. Edinburgh Math. Soc., 37, 271-300, (1994) · Zbl 0809.42018
[14] Lawton, W. M., Tight frames of compactly supported affine wavelets, J. Math. Phys., 31, 1898-1901, (Aug. 1990)
[15] Lemarié-Rieusset, P. G., Ondelettes à localisation exponentielle, J. Math. Pures Appl., 67, 227-236, (1988) · Zbl 0758.42020
[16] Lemarié-Rieusset, P. G., Existence de fonction-Père pour LES ondelettes à support compact, C. R. Acad. Sci. Paris, Sér. I, 314, 17-19, (1992) · Zbl 0752.42017
[17] Lemarié-Rieusset, P. G., Projecteurs invariants, matrices de dilation, ondelettes de dimensionn, Rev. Mat. Iberoameri., 10, 283-347, (1994) · Zbl 0807.42025
[18] Y. Meyer, 1990, Ondelettes et Opérateurs, I, II, Hermann, Paris
[19] A. Ron, Z. Shen, Affine Systems inL_{2}^{d}): The Analysis of the Analysis Operator, J. Funct. Anal. · Zbl 0891.42018
[20] X. Wang, 1995, Washington University
[21] E. Hernández, G. Weiss, 1996, A First Course in Wavelets, CRC Press, Boca Raton
[22] Wellend, G. V.; Lundberg, M., Construction of compactp, Constr. Approx., 9, 347-370, (1993) · Zbl 0784.42026
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