  We give a characterization of a class of band-limited wavelets of
$L^2({\mathbb R})$ and show that none of these wavelets come from a
multiresolution analysis (MRA). For each $n\geq 2$, we construct a subset $S_n$
of ${\mathbb R}$ which is symmetric with respect to the origin. We give
necessary and sufficient conditions on a function $\psi\in L^2({\mathbb R})$
with supp $\hat\psi\subseteq S_n$ to be an orthonormal wavelet. This result
generalizes the characterization of a class of wavelets of E. Hern\'andez and
G. Weiss. The dimension functions associated with these wavelets are also
computed explicitly. Starting from the wavelets we have constructed, we are
able to construct examples of wavelets in each of the equivalence classes of
wavelets defined by E. Weber.
