On 𝜇-statistical convergence

Bilalov, B.; Sadigova, S.

doi:10.1090/S0002-9939-2015-12528-2

On $\mu$-statistical convergence
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by B. T. Bilalov and S. R. Sadigova PDF

Proc. Amer. Math. Soc. 143 (2015), 3869-3878 Request permission

Abstract:

The concept of $\mu$-statistical convergence at a point for measurable functions in measurable space with a measure is introduced in this work. This concept is a generalization of a similar idea about the sequence of numbers. We also introduce the concept of $\mu$-statistical fundamentality at a point, and the equivalence of these two concepts is proved. The concept of $\mu$-statistical convergence at a point generalizes the usual one of the limit of a function at a point.

References

H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244 (1952) (French). MR 48548, DOI 10.4064/cm-2-3-4-241-244
T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), no. 2, 139–150 (English, with Russian summary). MR 587239
I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375. MR 104946, DOI 10.2307/2308747
J. A. Fridy, On statistical convergence, Analysis 5 (1985), no. 4, 301–313. MR 816582, DOI 10.1524/anly.1985.5.4.301
J. A. Fridy, Statistical limit points, Proc. Amer. Math. Soc. 118 (1993), no. 4, 1187–1192. MR 1181163, DOI 10.1090/S0002-9939-1993-1181163-6
A. Zygmund, Trigonometric series, Vol. II, Cambridge Univ. Press, London–New York, 1979.
J. S. Connor, The statistical and strong $p$-Cesàro convergence of sequences, Analysis 8 (1988), no. 1-2, 47–63. MR 954458, DOI 10.1524/anly.1988.8.12.47
Jeff Connor, $R$-type summability methods, Cauchy criteria, $P$-sets and statistical convergence, Proc. Amer. Math. Soc. 115 (1992), no. 2, 319–327. MR 1095221, DOI 10.1090/S0002-9939-1992-1095221-7
J. A. Fridy and H. I. Miller, A matrix characterization of statistical convergence, Analysis 11 (1991), no. 1, 59–66. MR 1113068, DOI 10.1524/anly.1991.11.1.59
I. J. Maddox, Statistical convergence in a locally convex space, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 1, 141–145. MR 938459, DOI 10.1017/S0305004100065312
D. Rath and B. C. Tripathy, On statistically convergent and statistically Cauchy sequences, Indian J. Pure Appl. Math. 25 (1994), no. 4, 381–386. MR 1272809
M. Mačaj and T. Šalát, Statistical convergence of subsequences of a given sequence, Math. Bohem. 126 (2001), no. 1, 191–208. MR 1826482
Jeff Connor and K.-G. Grosse-Erdmann, Sequential definitions of continuity for real functions, Rocky Mountain J. Math. 33 (2003), no. 1, 93–121. MR 1994482, DOI 10.1216/rmjm/1181069988
John A. Fridy and Mohammad K. Khan, Tauberian theorems via statistical convergence, J. Math. Anal. Appl. 228 (1998), no. 1, 73–95. MR 1659877, DOI 10.1006/jmaa.1998.6118
J. A. Fridy and H. I. Miller, A matrix characterization of statistical convergence, Analysis 11 (1991), no. 1, 59–66. MR 1113068, DOI 10.1524/anly.1991.11.1.59
Anindita Basu and P. D. Srivastava, Statistical convergence on composite vector valued sequence space, J. Math. Appl. 29 (2007), 75–90. MR 2345116
A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2002), no. 1, 129–138. MR 1911352, DOI 10.1216/rmjm/1030539612
Akif D. Gadjiev and Arash M. Ghorbanalizadeh, On the $A$-statistical approximation by sequences of $k$-positive linear operators, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 31 (2009), 41–52. MR 2675460
A. D. Gadjiev, Simultaneous statistical approximation of analytic functions and their derivatives by $k$-positive linear operators, Azerb. J. Math. 1 (2011), no. 1, 57–66. MR 2776097