On the q -Convolution on the Line |
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Authors: | Carnovale |
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Institution: | Dipartimento di Matematica Pura ed Applicata Università degli Studi di Padova via Belzoni 7 35131 Padova Italy carnoval@math.unipd.it, IT
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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. |
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