Band-limited wavelets beyond Gevrey regularity

It is known that a smooth function of exponential decay at infinity can not be an orthonormal wavelet. Dziubański and Hernández constructed smooth orthonormal wavelets of Gevrey type subexponential decay. We weaken the Gevrey type decay and construct orthonormal wavelets of subexponential decay related to the so-called extended Gevrey classes. The virtue of our construction is that prescise asymptotics of functions from such classes can be given in terms of the Lambert $W$ function.