1999/09/30 by G. Carnovale, T. H. Koornwinder
Mathematics · #math.CA #math.QA #msc:33D80 #msc:33D15 #msc:42A85 #msc:17B37
published as Methods Appl. Anal. 7 (2000), 705-726 · 31 pages; many small corrections; accepted by Methods and Applications of Analysis
arxiv created 2000/10/23 · arxiv updated 2009/11/30
In this paper we study a q-analogue of the convolution product on the line in detail. A convolution product on the braided line was defined algebraically by Kempf and Majid. We adapt their definition in order to give an analytic definition for the q-convolution and we study convergence extensively. Since the braided line is commutative as an algebra, all results can be viewed both as results in classical q-analysis and in braided algebra. We define various classes of functions on which the convolution is well-defined and we show that they are algebras under the defined product. One particularly nice family of algebras, a decreasing chain depending on a parameter running through (0,1], turns out to have 1/2 as the critical parameter value above which the algebras are commutative. Morerover, the commutative algebras in this family are precisely the algebras in which each function is determined by its q-moments. We also treat the relationship between q-convolution and q-Fourier transform. Finally, in the Appendix, we show an equivalence between the existence of an analytic continuation of a function defined on a q-lattice, and the behaviour of its q-derivatives.