  It is known that a smooth function of exponential decay at infinity can not
be an orthonormal wavelet. Dziuba\'nski and Hern\'andez 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.
