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August  2022, 5(3): 231-239. doi: 10.3934/mfc.2021032

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

Michele Campiti

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

Received  August 2021 Revised  October 2021 Published  August 2022 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.

Keywords: Korovkin approximation, approximation of vector-valued functions, approximation of set-valued functions, convexity-monotone operators, linear positive continuous operators.
Mathematics Subject Classification: Primary: 41A65, 41A36; Secondary: 41A25, 41A63.
Citation: Michele Campiti. Korovkin-type approximation of set-valued and vector-valued functions. Mathematical Foundations of Computing, 2022, 5 (3) : 231-239. 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.

[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.

show all references

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.

[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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