  In this paper we study a q-analogue of the convolution product on the line in
detail. A convolution product on the braided line was defined algebraically by
Kempf and Majid. We adapt their definition in order to give an analytic
definition for the q-convolution and we study convergence extensively. Since
the braided line is commutative as an algebra, all results can be viewed both
as results in classical q-analysis and in braided algebra. We define various
classes of functions on which the convolution is well-defined and we show that
they are algebras under the defined product. One particularly nice family of
algebras, a decreasing chain depending on a parameter running through (0,1],
turns out to have 1/2 as the critical parameter value above which the algebras
are commutative. Morerover, the commutative algebras in this family are
precisely the algebras in which each function is determined by its q-moments.
We also treat the relationship between q-convolution and q-Fourier transform.
Finally, in the Appendix, we show an equivalence between the existence of an
analytic continuation of a function defined on a q-lattice, and the behaviour
of its q-derivatives.
