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Korovkin-type approximation of set-valued and vector-valued functions

Work performed under the auspices of G.N.A.M.P.A. (I.N.d.A.M.) and the UMI Group TAA "Approximation Theory and Applications"

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  • We establish some general Korovkin-type results in cones of set-valued functions and in spaces of vector-valued functions. These results constitute a meaningful extension of the preceding ones.

    Mathematics Subject Classification: Primary: 41A65, 41A36; Secondary: 41A25, 41A63.

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  • [1] F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and its Applications, De Gruyter Studies in Mathematics, 17, Berlin-Heidelberg-New York, 1994. doi: 10.1515/9783110884586.
    [2] F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Rașa, Markov Operators, Positive Semigroups and Approximation Processes, , De Gruyter Studies in Mathematics, 61, Berlin-Munich-Boston, 2014. doi: 10.1515/9783110366976.
    [3] M. Campiti, A Korovkin-type theorem for set-valued Hausdorff continuous functions, Matematiche (Catania), 42 (1987), 29–35.
    [4] M. Campiti, Approximation of continuous set-valued functions in Fréchet spaces I, Anal. Numér. Théor. Approx., 20 (1991), 15–23.
    [5] M. Campiti, Approximation of continuous set-valued functions in Fréchet spaces II, Anal. Numér. Théor. Approx., 20 (1991), 25–38.
    [6] M. Campiti, Korovkin theorems for vector-valued continuous functions, in Approximation Theory, Spline Functions and Applications (Internat. Conf., Maratea, May 1991), 293–302, Nato Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 356, Kluwer Acad. Publ., Dordrecht, 1992.
    [7] M. Campiti, Convergence of nets of monotone operators between cones of set-valued functions, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 126 (1992), 39–54.
    [8] M. Campiti, Convexity-monotone operators in Korovkin theory, Rend. Circ. Mat. Palermo, 33 (1993), 229–238.
    [9] M. Campiti, Korovkin-type approximation in spaces of vector-valued and set-valued functions, Appl. Anal., 98 (2019), 2486–2496. doi: 10.1080/00036811.2018.1463522.
    [10] M. Campiti, On the Korovkin-type approximation of set-valued continuous functions, Constr. Math. Anal., 4 (2021), 119–134. doi: 10.33205/cma. 863145.
    [11] K. Keimel and W. Roth, A Korovkin type approximation theorem for set-valued functions, Proc. Amer. Math. Soc., 104 (1988), 819–824. doi: 10.1090/S0002-9939-1988-0964863-8.
    [12] K. Keimel and W. Roth, Ordered Cones and Approximation, Lecture Notes in Mathematics, 1517, Springer-Verlag Berlin Heidelberg, 1992. doi: 10.1007/BFb0089190.
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