Abstract
The classical summability theory can not be used in the topological spaces as it needs addition operator. Recently some authors have studied the summability theory in the topological spaces by assuming the topological space to have a group structure or a linear structure or introducing some summability methods those do not need a linear structure in the topological space as statistical convergence and distributional convergence. In the present paper we introduce a new concept of density and we study the summability theory in an arbitrary Hausdorff space by introducing a new type of statistical convergence and distirbutional convergence via Abel method that is a sequence-to-function transformation. Moreover we give a Bochner integral representation of \(Abel\)-summability in the Banach spaces.
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References
Boos, J.: Classical and Modern Methods in Summability. Oxford University Press, Oxford (2000)
Buck, R.C.: Generalized asymptotic density. Am. J. Math. 75, 335–346 (1953)
Buck, R.C.: The measure theoretic approach to density. Am. J. Math. 68, 560–580 (1946)
Cakalli, H., Khan, M.K.: Summability in topological spaces. Appl. Math. Lett. 24(3), 348–352 (2011)
Cakalli, H.: Lacunary statistical convergence in topological groups. Indian J. Pure Appl. Math. 26(2), 113–119 (1995)
Cakalli, H.: On statistical convergence in topological groups. Pure Appl. Math. Sci. 43(1–2), 27–31 (1996)
Cakalli, H., Albayrak, M.: New type continuities via Abel convergence. Sci. World J. 2014 (2014)
Connor, J.S.: The statistical and strong p-Cesàro convergence of sequences. Analysis 8(1–2), 47–63 (1988)
Di Maio, G., Kočinac, L.D.R.: Statistical convergence in topology. Topol. Appl. 156(1), 28–45 (2008)
Estrada, R., Vindas, J.: Distributional versions of Littlewood’s Tauberian theorem. Czech. Math. J. 63(2), 403–420 (2013)
Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)
Fridy, J.A.: On statistical convergence. Analysis 5(4), 301–313 (1985)
Fridy, J.A., Miller, H.I.: A matrix characterization of statistical convergence. Analysis 11(1), 59–66 (1991)
Osikiewicz, J.A.: Summability of spliced sequences. Rocky Mt. J. Math. 35(3), 977–996 (2005)
Powell, R.E., Shah, S.M.: Summability Theory and Its Applications. Prentice-Hall of India, New Delhi (1988)
Prullage, D.L.: Summability in topological groups. Math. Z. 96, 259–278 (1967)
Prullage, D.L.: Summability in topological groups. II. Math. Z. 103, 129–138 (1968)
Prullage, D.L.: Summability in topological groups. IV. Convergence fields. Tôhoku Math. J. 2(21), 159–169 (1969)
Prullage, D.L.: Summability in topological groups. III. Metric properties. J. Anal. Math. 22, 221–231 (1969)
Šalát, T.: On statistically convergent sequences of real numbers. Math. Slovaca 30(2), 139–150 (1980)
Steinhaus, H.: Some remarks on the generalizations of the notion of limit. Prace Mat. Fiz. 22, 121–134 (1911) (in Polish)
Unver, M., Khan, M.K., Orhan, C.: \(A\)-distributional summability in topological spaces. Positivity 18, 131–145 (2014)
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Communicated by G. Teschl.
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Ünver, M. Abel summability in topological spaces. Monatsh Math 178, 633–643 (2015). https://doi.org/10.1007/s00605-014-0717-0
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DOI: https://doi.org/10.1007/s00605-014-0717-0