  I continue the investigation of a q-analogue of the convolution on the line
started in a joint work with Koornwinder and based on a formal definition due
to Kempf and Majid.
  Two different ways of approximating functions by means of the convolution and
convolution of delta functions are introduced. A new family of functions that
forms an increasing chain of algebras depending on a parameter s>0 is
constructed. The value of the parameter for which the mentioned algebras are
well behaved, commutative and unital is found. In particular a privileged
algebra of functions belonging to the above family is shown to be the quotient
of an algebra studied in the previous article modulo the kernel of a q-analogue
of the Fourier transform. This result has an analytic interpretation in terms
of analytic functions whose q-moments have a particular behaviour. The same
result makes it possible to extend results on invertibility of the q-Fourier
transform due to Koornwinder. A few results on invertibility of functions with
respect to the q-convolution are also obtained and they are related to solving
certain simple linear q-difference equations with polynomial coefficients.
