Abstract
The ordinary Riemann integral can be regarded as an extension of the integral of a continuous function to a larger space. The extension procedure given below generalizes to other positive linear functionals and operators. It yields some useful characterizations of sets of uniform convergence for such functionals and operators. There are applications to the approximate solution of integral equations by means of numerical integration. The same kind of analysis also leads to generalizations of Korovkin’s theorem on the convergence of positive linear operators from its original setting in C [0, 1] to an arbitrary partially ordered Banach space setting.
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