# American Institute of Mathematical Sciences

doi: 10.3934/mfc.2021032
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## Korovkin-type approximation of set-valued and vector-valued functions

 Department of Mathematics and Physics "E. De Giorgi", University of Salento, Campus Ecotekne, 73100 Lecce, Italy

Received  August 2021 Revised  October 2021 Early access November 2021

Fund Project: 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"

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.

Citation: Michele Campiti. Korovkin-type approximation of set-valued and vector-valued functions. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021032
##### References:
 [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.  Google Scholar [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.  Google Scholar [3] M. Campiti, A Korovkin-type theorem for set-valued Hausdorff continuous functions, Matematiche (Catania), 42 (1987), 29–35.  Google Scholar [4] M. Campiti, Approximation of continuous set-valued functions in Fréchet spaces I, Anal. Numér. Théor. Approx., 20 (1991), 15–23.  Google Scholar [5] M. Campiti, Approximation of continuous set-valued functions in Fréchet spaces II, Anal. Numér. Théor. Approx., 20 (1991), 25–38.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] M. Campiti, Convexity-monotone operators in Korovkin theory, Rend. Circ. Mat. Palermo, 33 (1993), 229–238.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] K. Keimel and W. Roth, Ordered Cones and Approximation, Lecture Notes in Mathematics, 1517, Springer-Verlag Berlin Heidelberg, 1992. doi: 10.1007/BFb0089190.  Google Scholar

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##### References:
 [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.  Google Scholar [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.  Google Scholar [3] M. Campiti, A Korovkin-type theorem for set-valued Hausdorff continuous functions, Matematiche (Catania), 42 (1987), 29–35.  Google Scholar [4] M. Campiti, Approximation of continuous set-valued functions in Fréchet spaces I, Anal. Numér. Théor. Approx., 20 (1991), 15–23.  Google Scholar [5] M. Campiti, Approximation of continuous set-valued functions in Fréchet spaces II, Anal. Numér. Théor. Approx., 20 (1991), 25–38.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] M. Campiti, Convexity-monotone operators in Korovkin theory, Rend. Circ. Mat. Palermo, 33 (1993), 229–238.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] K. Keimel and W. Roth, Ordered Cones and Approximation, Lecture Notes in Mathematics, 1517, Springer-Verlag Berlin Heidelberg, 1992. doi: 10.1007/BFb0089190.  Google Scholar
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