Canadian Mathematical Society
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Abstract view

Band-limited wavelets with subexponential decay

 Printed: Dec 1998
  • Jacek Dziubański
  • Eugenio Hernández
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax   PDF   PostScript  


It is well known that the compactly supported wavelets cannot belong to the class $C^\infty({\bf R})\cap L^2({\bf R})$. This is also true for wavelets with exponential decay. We show that one can construct wavelets in the class $C^\infty({\bf R})\cap L^2({\bf R})$ 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 Lemari\'e-Meyer wavelets \cite{LM} that is found in \cite{BSW} so that we obtain band-limited, $C^\infty$-wavelets on $\bf R$ that have subexponential decay, that is, for every $0<\varepsilon<1$, there exits $C_\varepsilon>0$ such that $|\psi(x)|\leq C_\varepsilon e^{-|x|^{1-\varepsilon}}$, $x\in\bf R$. Moreover, all of its derivatives have also subexponential decay. The proof is constructive and uses the Gevrey classes of functions.
Keywords: Wavelet, Gevrey classes, subexponential decay Wavelet, Gevrey classes, subexponential decay
MSC Classifications: 42C15 show english descriptions General harmonic expansions, frames 42C15 - General harmonic expansions, frames

© Canadian Mathematical Society, 2014 :