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Band-Limited Wavelets with Subexponential Decay

Published online by Cambridge University Press:  20 November 2018

Jacek Dziubański
Affiliation:
Instytut Matematyczny Uniwersytet Wroclawski 50-384 Wroclaw Poland, e-mail: jdziuban@math.uni.wroc.pl
Eugenio Hernández
Affiliation:
Departamento Matemáticas Universidad Autónoma de Madrid 28049 Madrid Spain, e-mail: eugenio.hernandez@uam.es
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Abstract

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It is well known that the compactly supported wavelets cannot belong to the class ${{C}^{\infty }}\,(\text{R})\,\cap \,{{L}^{2}}\,(R)$. This is also true for wavelets with exponential decay. We show that one can construct wavelets in the class ${{C}^{\infty }}\,(\text{R})\,\cap \,{{L}^{2}}\,(R)$ that are “almost” of exponential decay and, moreover, they are band-limited. We do this by showing that we can adapt the construction of the Lemarié-Meyer wavelets $[\text{LM }\!\!]\!\!\text{ }$ that is found in $[\text{BSW}]$ so that we obtain band-limited, ${{C}^{\infty }}$-wavelets on $R$ that have subexponential decay, that is, for every $0<\varepsilon <1$, there exits ${{C}_{\in }}\,>\,0$ such that $|\psi (x)|\le {{C}_{\varepsilon }}{{e}^{-|x{{|}^{1-\varepsilon }}}}$, $x\in \text{R}$. Moreover, all of its derivatives have also subexponential decay. The proof is constructive and uses the Gevrey classes of functions.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1998

References

[AWW] Auscher, P., Weiss, G. and Wickerhauser, M. V., Local sine and cosine basis of Coifman and Meyer and the construction of smooth wavelets. In:Wavelets: A tutorial in theory and applications, (ed. C.K. Chui), Academic Press, (1992), 237256.Google Scholar
[BSW] Bonami, A., Soria, F. and Weiss, G., Band-limited wavelets. J. Geom. Anal. (6) 3 (1993), 544578.Google Scholar
[Da] Daubechies, I., Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (1988), 909996.Google Scholar
[Ho1] Hörmander, L., The analysis of linear partial differential operators I. Springer-Verlag, 1983.Google Scholar
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[HW] Hernández, E. and Weiss, G., A first course on wavelets. CRC Press, 1996.CrossRefGoogle Scholar
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[LM] Lemarié, P. G. and Meyer, Y., Ondelettes et bases hilbertiennes, Rev. Mat. Iberoamericana 2 (1986), 118.Google Scholar
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