Approximation Theory and Functional Analysis on Time Scales

Here we start by proving the Riesz representation theorem for positive linear functionals on the space of continuous functions over a time scale. Then we prove further properties for the related Riemann–Stieltjes integral on time scales and we prove the related Hölder’s inequality. Next we prove the Hölder’s inequality for general positive linear functionals on time scales. We introduce basic concepts of Approximation theory on time scales and we discuss some limitations of the modulus of continuity there. Next we prove the famous Korovkin theorem on time scales, regarding the approximation of unit operator by sequences of positive linear operators on the space of continuity functions defined on a compact interval of a time scale. Then we produce several Shisha–Mond type inequalities related to Korovkin’s theorem, putting the convergence of positive linear operators and positive linear functionals in a quantitative form and giving rates of convergence, all operating on Lipschitz functions on a time scale. At the end we present an example of a concrete and genuine positive linear operator on time scales and we give its approximation and interpolation properties over continuous functions. AMS Subject Classifications: 39A12, 41A17, 41A25, 41A36, 47A58, 47A67, 93C70.