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.
Supported by Atomic Energy Commission Contract No. AT (45-1)-1947.
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References
Anselone, P.M.: Uniform Approximation Theory for Integral Equations with Discontinuous Kernels. SIAM J. Num. Anal. 4 (1967), 245–253.
Anselone, P.M.: Collectively Compact Approximations of Integral Operators with Discontinuous Kernels. J. Math. Anal. Appl. 22 (1968), 582–590.
Cheney, E.W.: Introduction to Approximation Theory. McGraw-Hill Book Co., New York, 1966.
James, R. L.: Convergence of Positive Operators. Ph.D. Thesis, Oregon State University, 1970.
Szegö, G.: Orthogonal Polynomials (revised edition), American Mathematical Society Colloquium Publication XXIII, New York, 1959.
Wulbert, D.E., Convergence of Operators and Korovkin’s Theorem, J.Approx. Theory, 1 (1968), 381–390.
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Anselone, P.M. (1970). Abstract Riemann Integrals, Monotone Approximations, and Generalizations of Korovkin’s Theorem. In: Collatz, L., Meinardus, G., Unger, H., Werner, H. (eds) Iterationsverfahren Numerische Mathematik Approximationstheorie. Internationale Schriftenreihe zur Numerischen Mathematik / International Series of Numerical Mathematics / Série Internationale D’Analyse Numérique, vol 15. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5833-5_13
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DOI: https://doi.org/10.1007/978-3-0348-5833-5_13
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