Abstract
The main purpose of this paper is to give a procedure to “mollify” the low-pass filters of a large number ofMinimally Supported Frequency (MSF) wavelets so that the smoother functions obtained in this way are also low-pass filters for an MRA. Hence, we are able to approximate (in the L2-norm) MSF wavelets by wavelets with any desired degree of smoothness on the Fourier transform side. Although the MSF wavelets we consider are bandlimited, this may not be true for their smooth approximations. This phenomena is related to the invariant cycles under the transformation x ↦2x (mod2π). We also give a characterization of all low-pass filters for MSF wavelets. Throughout the paper new and interesting examples of wavelets are described.
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References
Auscher, P.Solutions to two problems on wavelets. J. Geom. Anal., to appear.
Auscher, P., Wickerhauser, M. V., and Weiss, G. (1992).Local sine and cosine bases of Coifman and Meyer and the construction of smooth wavelets. Wavelets: a Tutorial in Theory and Applications (C.K. Chui, ed.). Academic Press, New York, 237–256.
Bonami, A., Soria, F., and Weiss, G. (1993).Band-limited wavelets.J. Geom. Anal. 3, 543–578.
Cohen, A. (1990).Ondelettes, analyse multirésolutions et filters miroirs en quadrature.Ann. Inst. H. Poincaré Anal. Non-Lin. 7, 439–459.
Cohen, A. (1992).Ondelettes et Traitement Numérique du Signal. Masson, Paris.
Daubechies, I. (1992).Ten lectures on wavelets.CBS-NSF Reg. Conf. Appl. Math. 61. Society for Industrial and Applied Mathematics, Philadelphia, PA.
Fang, X., and Wang, X. (1996).Construction of minimally-supported-frequency wavelets.The J. Fourier Anal. Appl.,2, 315–327.
Ha, Y.-H., Kang, H., Lee, J., and Seo, J. (1994).Unimodular wavelets for L 2 and the Hardy space H 2.Michigan Math. J. 41, 723–736.
Hernández, E., and Weiss, G. (1996).A First Course on Wavelets. CRC Press, Boca Raton, FL.
Hernández, E., Wang, X., and Weiss, G. (1995).Smoothing minimally supported frequency (MSF) wavelets: Part I.The J. Fourier Anal. Appl.,2, 329–340.
Lemarié, P.G., and Meyer, Y. (1986).Ondelettes et bases Hilbertiennes.Rev. Mat. Iberoamericana 2, 1–18.
Meyer, Y. (1990).Ondelettes et Opérateurs. Herman, Paris.
Wang, X. (1995).The study of wavelets from the properties of their Fourier transforms. Ph.D. Thesis. Washington University.
Wickerhauser, M. V. (1994).Adapted Wavelet Analysis from Theory to Software. A.K. Peters.
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Hernández, E., Wang, X. & Weiss, G. Smoothing minimally supported frequency wavelets: Part II. The Journal of Fourier Analysis and Applications 3, 23–41 (1997). https://doi.org/10.1007/BF02647945
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DOI: https://doi.org/10.1007/BF02647945