Basic Properties of Statistical Epi-Convergence

Year 2020, Volume: 20 Issue: 6, 968 - 974, 31.12.2020

Şükrü Tortop

https://doi.org/10.35414/akufemubid.819410

Abstract

In this paper, we give some basic properties in order to use statistical epi-convergence more efficiently in future studies. Such situations are studied: Uniform statistical convergence of sequence of functions, statistical epi-limit of compound of sequence of functions, statistical epi-limit of the sum of sequence of functions, the property of epi-limit function if the sequence of functions are lower semi-continuous and the convexity of epi-limit function if each function in the sequence is convex.

Keywords

Epi-convergence, Statistical Convergence, Epigraph, Sequence of Functions

İstatistiksel Epi-Yakınsaklık ile İlgili Temel Özellikler

Abstract

Bu çalışmada, istatistiksel epi-yakınsaklığın sonraki çalışmalarda daha verimli kullanılabilmesi için bazı temel özelliklere yer verildi. Bir fonksiyon dizisinin düzgün istatistiksel yakınsaklık durumu, fonksiyon dizilerinin bileşkesinin istatistiksel epi-limiti, fonksiyon dizilerinin toplamının istatistiksel epi-limiti, fonksiyon dizisinin alttan yarı sürekli olması halinde epi-limit fonksiyonunun özelliği ve fonksiyon dizisindeki her bir fonksiyonun konveks olması halinde epi-limit fonksiyonunun konveksliği gibi durumlar çalışıldı.

Keywords

Epi-Yakınsaklık, İstatistiksel Yakınsaklık, Epigraf, Fonksiyon Dizileri

References

• Anastassiou, A. G. and Duman, O., 2011. Towards Intelligent Modeling: Statistical Approximation Theory, vol.14, Berlin.
• Attouch, H., 1977. Convergence de fonctions convexes, de sous-differentiels et semi-groupes. Comptes Rendus de lAcademie des Sciences de Paris, 284, 539-542.
• Caserta, A. and Koc ̆inac, Lj. D. R., 2012. On statistical exhaustiveness. Applied Mathematics Letters, 25, 1447-1451.
• Di Maio, G. and Koc ̆inac, Lj. D. R., 2008. Statistical convergence in topology. Topology and its Applications, 156, 28-45.
• Duman, O. and Orhan, C., 2004. μ -statistically convergent function sequences. Czechoslovak Mathematical Journal, 54 (129)(2), 413-422.
• Fast, H., 1951. Sur la convergence statistique. Colloquium Mathematicum, 2, 241–244.
• Fridy, J. A., 1993. Statistical limit points. Proceedings of the American Mathematical Society, 118 (4), 1182–1192.
• Fridy, J. A. and Orhan, C., 1997. Statistical limit superior and limit inferior. Proceedings of the American Mathematical Society, 125, 3625–3631.
• Gökhan, A. and Güngör, M., 2002. On pointwise statistical convergence. Indian Journal of Pure and Applied Mathematics, 33 (9), 1379-1384.
• Güngör, M. and Gökhan, A., 2005. On uniform statistical convergence. International Journal of Pure and Applied Mathematics, 19 (1), 17–24.
• Joly, J.-L., 1973. Une famille de topologies sur lensemble des fonctions convexes pour lesquelles la polarite est bicontinue. Journal de Mathematiques Pures et Appliquees, 52, 421–441.
• Kuratowski, C., 1958. Topologie, vol.I, PWN, Warszawa.
• Maso, G. D., 1993. An introduction to Γ-convergence, vol.8, Boston.
• McLinden, L. and Bergstrom, R., 1981. Preservation of convergence of sets and functions in finite dimensions. Transactions of the American Mathematical Society, 268, 127–142.
• Mosco, U., 1969. Convergence of convex sets and of solutions of variational inequalities. Advances in Mathematics, 3, 510–585.
• Niven, I. and Zuckerman, H. S., 1980. An Introduction to the Theory of Numbers, New York.
• Rockafellar, R.T. and Wets, R.J-B., 2009. Variational Analysis, Berlin.
• S ̆ala't, T., 1980. On statistically convergent sequences of real numbers. Mathematica Slovaca, 30, 139-150.
• Salinetti, G. and Wets, R.J-B., 1977. On the relation between two types of convergence for convex functions. Journal of Mathematical Analysis and Applications, 60, 211–226.
• Schoenberg, I.J.:, 1959. The integrability of certain functions and related summability methods. American Mathematical Monthly, 66, 361-375.
• Steinhaus, H., 1951. Sur la convergence ordinaire et la convergence asymptotique. Colloquium Mathematicum, 2, 73-74.
• Talo, Ö., Sever, Y. and Başar, F., 2016. On statistically convergent sequences of closed sets. Filomat, 30 (6), 1497-1509.
• Wets, R.J-B., 1980. Convergence of convex functions, variational inequalities and convex optimization problems, New York.
• Wijsman, R. A., 1964. Convergence of sequences of convex sets, cones and functions. Bulletin of American Mathematical Society, 70, 186-188.
• Wijsman, R. A., 1966. Convergence of sequences of convex sets, cones and functions II. Transactions of the American Mathematical Society, 123, 32-45.
• Zygmund, A., 1979. Trigonometric Series, Cambridge University Press, Cambridge, UK.