On the q-convolution on the line

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.