Band-Limited Wavelets with Subexponential Decay

Dziubański, Jacek ; Hernández, Eugenio

Canadian mathematical bulletin, Tome 41 (1998) no. 4, pp. 398-403

Résumé

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.

Détail

DOI : 10.4153/CMB-1998-053-8

Mots-clés : 42C15, Wavelet, Gevrey classes, subexponential decay