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The convolution on time scales. (English) Zbl 1155.39010
The authors suggest a new definition of convolution for functions on time scales. This is done by utilizing a certain shift-operator on time scales, defined as a solution of a partial dynamic equation. The existence of this operator is assured by quite technical restrictive conditions involving power series. As for concrete examples the authors return to the cases where group actions on the relevant time scales are present. Especially the case of $q$-difference equations is considered.

##### MSC:
 39A12 Discrete version of topics in analysis 44A35 Convolution (integral transforms) 39A13 Difference equations, scaling ($q$-differences) 44A10 Laplace transform
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##### References:
 [1] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction With Applications, Birkhäuser, Boston, Mass, USA, 2001. · Zbl 0978.39001 [2] M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. · Zbl 1025.34001 [3] S. Hilger, “Special functions, Laplace and Fourier transform on measure chains,” Dynamic Systems and Applications, vol. 8, no. 3-4, pp. 471-488, 1999, special issue on “Discrete and Continuous Hamiltonian Systems” edited by R. P. Agarwal and M. Bohner. · Zbl 0943.39006 [4] M. Bohner and A. Peterson, “Laplace transform and Z-transform: unification and extension,” Methods and Applications of Analysis, vol. 9, no. 1, pp. 151-157, 2002, preprint in Ulmer Seminare 6. · Zbl 1031.44001 [5] M. Bohner and G. Sh. Guseinov, “Partial differentiation on time scales,” Dynamic Systems and Applications, vol. 13, no. 3-4, pp. 351-379, 2004. · Zbl 1090.26004 [6] R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,” Results in Mathematics, vol. 35, no. 1-2, pp. 3-22, 1999. · Zbl 0927.39003 · doi:10.1007/BF03322019 [7] M. Bohner and D. A. Lutz, “Asymptotic expansions and analytic dynamic equations,” ZAMM Zeitschrift für Angewandte Mathematik und Mechanik, vol. 86, no. 1, pp. 37-45, 2006. · Zbl 1092.39002 · doi:10.1002/zamm.200410219 [8] V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, New York, NY, USA, 2002. · Zbl 0986.05001