Band-Limited Wavelets with Subexponential Decay

It is well known that the compactly supported wavelets cannot belong to the class C∞(R) ∩ L(R). This is also true for wavelets with exponential decay. We show that one can construct wavelets in the class C∞(R)∩L2(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 [LM] that is found in [BSW] so that we obtain band-limited, C∞-wavelets on R that have subexponential decay, that is, for every 0 < ε < 1, there exits Cε > 0 such that |ψ(x)| ≤ Cε e−|x| 1−ε , x ∈ R. Moreover, all of its derivatives have also subexponential decay. The proof is constructive and uses the Gevrey classes of functions. ∗Research supported by grant 2 P301 052 07 from KNB, Poland. †Research supported by grant PB94-149 from DGICYT (Ministerio de Educación y Ciencia, Spain) and by a grant from Southwestern Bell Telephone Company. Math Subject Classification. Primary 42C15