The investigation of a q-analogue of the convolution on the line, started in conjunction with Koornwinder, is continued, 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 is shown to be the quotient of an algebra studied in a previous paper 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.
On the q-convolution on the line
CARNOVALE, GIOVANNA
2002
Abstract
The investigation of a q-analogue of the convolution on the line, started in conjunction with Koornwinder, is continued, 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 is shown to be the quotient of an algebra studied in a previous paper 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.Pubblicazioni consigliate
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