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Basic Properties of Statistical Epi-Convergence

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

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.

İstatistiksel Epi-Yakınsaklık ile İlgili Temel Özellikler

Year 2020, Volume: 20 Issue: 6, 968 - 974, 31.12.2020
https://doi.org/10.35414/akufemubid.819410

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.

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.
There are 26 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Şükrü Tortop 0000-0001-5342-7612

Publication Date December 31, 2020
Submission Date November 1, 2020
Published in Issue Year 2020 Volume: 20 Issue: 6

Cite

APA Tortop, Ş. (2020). Basic Properties of Statistical Epi-Convergence. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, 20(6), 968-974. https://doi.org/10.35414/akufemubid.819410
AMA Tortop Ş. Basic Properties of Statistical Epi-Convergence. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. December 2020;20(6):968-974. doi:10.35414/akufemubid.819410
Chicago Tortop, Şükrü. “Basic Properties of Statistical Epi-Convergence”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 20, no. 6 (December 2020): 968-74. https://doi.org/10.35414/akufemubid.819410.
EndNote Tortop Ş (December 1, 2020) Basic Properties of Statistical Epi-Convergence. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 20 6 968–974.
IEEE Ş. Tortop, “Basic Properties of Statistical Epi-Convergence”, Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 20, no. 6, pp. 968–974, 2020, doi: 10.35414/akufemubid.819410.
ISNAD Tortop, Şükrü. “Basic Properties of Statistical Epi-Convergence”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 20/6 (December 2020), 968-974. https://doi.org/10.35414/akufemubid.819410.
JAMA Tortop Ş. Basic Properties of Statistical Epi-Convergence. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2020;20:968–974.
MLA Tortop, Şükrü. “Basic Properties of Statistical Epi-Convergence”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 20, no. 6, 2020, pp. 968-74, doi:10.35414/akufemubid.819410.
Vancouver Tortop Ş. Basic Properties of Statistical Epi-Convergence. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2020;20(6):968-74.