  We generalize some widely used mother wavelets by means of the q-exponential
function $e_q^x \equiv [1+(1-q)x]^{1/(1-q)}$ ($q \in {\mathbb R}$, $e_1^x=e^x$)
that emerges from nonextensive statistical mechanics. Particularly, we define
extended versions of the mexican hat and the Morlet wavelets. We also introduce
new wavelets that are $q$-generalizations of the trigonometric functions. All
cases reduce to the usual ones as $q \to 1$. Within nonextensive statistical
mechanics, departures from unity of the entropic index q are expected in the
presence of long-range interactions, long-term memory, multi-fractal
structures, among others. Consistently the analysis of signals associated with
such features is hopefully improved by proper tuning of the value of q. We
exemplify with the WTMM Method for mono- and multi-fractal self-affine signals.
