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.