Band-limited wavelets with subexponential decay

It is well known that the compactly supported wavelets cannot belong to the class C-infinity(R) boolean AND L-2(R). This is also true for wavelets with exponential decay. We show that one can construct wavelets in the class C-infinity(R)boolean AND 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 Lemarie-Meyer wavelets [LM] that is found in [BSW] so that we obtain band-limited, C-infinity-wavelets on R that have subexponential decay, that is, for every 0 < epsilon < 1, there exits C-epsilon > 0 such that I\psi(x)\ less than or equal to C(epsilon)e(-)\x\(1-epsilon), x is an element of R Moreover, all of its derivatives have also subexponential decay. The proof is constructive and uses the Gevrey classes of functions.