On the \(q\)-convolution on the line (Q1862745)

In this paper the author continues the investigation of a \(q\)-analogue of the convolution on the line, started in [\textit{G. Carnovale} and \textit{T. H. Koornwinder}, Methods Appl. Anal. 7, 705-726 (2000; Zbl 1004.33013)], with special attention to the approximation of functions by means of the convolution. A new space of functions that forms an increasing chain of algebras (with respect to the \(q\)-convolution), depending on a parameter \(s>0\), is constructed. For a special value of the parameter the corresponding algebra is commutative and unital, and it is shown to be the quotient of an algebra studied in the paper cited above 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 fast decreasing behavior and allows the extension of Koornwinder's inversion formula for the \(q\)-Fourier transform. A few results on the invertibility of functions with respect to the \(q\)-convolution are also obtained and they are applied to the solution of certain simple linear \(q\)-difference equations with polynomial coefficients.