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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Carnovale On the q -Convolution on the Line . Constr. Approx. 18, 309–341 (2002). https://doi.org/10.1007/s00365-001-0028-2
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DOI: https://doi.org/10.1007/s00365-001-0028-2
Key words
- q-Convolution
- q -Fourier transform
- q-Moment problem
- Determinacy of the q -moment problem
- Algebras of functions, Approximation of functions