On the Korovkin-type approximation of set-valued continuous functions

This paper is devoted to some Korovkin approximation results in cones of Hausdorff continuous set-valued functions and in spaces of vector valued functions. Some classical results are exposed in order to give a more complete treatment of the subject. New contributions are concerned both with the general theory than in particular with the so-called convexity monotone operators, which are considered in cones of set-valued function and also in spaces of vector-valued functions.

Keywords:

Korovkin approximation, approximation of vector-valued functions approximation of set-valued functions,

PDF

___

F. Altomare, M. Campiti: Korovkin-type Approximation Theory and Its Applications, De Gruyter Studies in Mathematics 17, Berlin-Heidelberg-New York, (1994).
F. Altomare, M. Cappelletti, V. Leonessa and I. Ra¸sa: Markov Operators, Positive Semigroups and Approximation Processes, De Gruyter Studies in Mathematics 61, Berlin-Munich-Boston, (2015).
H. Berens, G. G. Lorentz: Geometric theory of Korovkin sets, J. Approx. Theory, 15 (3) (1975), 161–189.
M. Campiti: A Korovkin-type theorem for set-valued Hausdorff continuous functions, Le Mathematiche, 42 (I–II) (1987), 29–35.
M. Campiti: Approximation of continuous set-valued functions in Fréchet spaces I, Rev. Anal. Numér. Théor. Approx., 20 (1–2) (1991), 15–23.
M. Campiti: Approximation of continuous set-valued functions in Fréchet spaces II, Rev. Anal. Numér. Théor. Approx., 20 (1–2) (1991), 24–38.
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.
M. Campiti: Convergence of nets of monotone operators between cones of set-valued functions, Atti dell’Accademia delle Scienze di Torino, 126 (1992), 39–54.
M. Campiti: Convexity-monotone operators in Korovkin theory, Rend. Circ. Mat. Palermo, 33 (1993), 229–238.
M. Campiti: Korovkin-type approximation in spaces of vector-valued and set-valued functions, Applicable Analysis, 98 (13) (2019), 2486–2496.
L. B. O. Ferguson, M. D. Rusk: Korovkin sets for an operator on a space of continuous functions, Pacific J. Math., 65 (2) (1976), 337–345.
W. Heping: Korovkin-type theorem and application, J. Approx. Theory, 132 (2005), 258–264.
K. Keimel, W. Roth: A Korovkin type approximation theorem for set-valued functions, Proc. Amer. Math. Soc., 104 (1988), 819–824.
K. Keimel, W. Roth: Ordered cones and approximation, Lecture Notes in Mathematics, 1517, Springer-Verlag Berlin Heidelberg, (1992).
N. I. Mahmudov: Korovkin-type theorems and applications, Cent. Eur. J. Math., 7 (2) (2009), 348–356.
T. Nishishiraho: Convergence of quasi-positive linear operators, Atti Sem. Mat. Fis. Univ. Modena, 29 (1991), 367–374.