Real Analysis Exchange

I-Convergence

Pavel Kostyrko, Władysław Wilczyński, and Tibor Šalát

Full-text: Open access

Abstract

In this paper we introduce and study the concept of ${\cal I}$-convergence of sequences in metric spaces, where ${\cal I}$ is an ideal of subsets of the set $\N$ of positive integers. We extend this concept to ${\cal I}$-convergence of sequence of real functions defined on a metric space and prove some basic properties of these concepts.

Article information

Source
Real Anal. Exchange, Volume 26, Number 2 (2000), 669-686.

Dates
First available in Project Euclid: 27 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.rae/1214571359

Mathematical Reviews number (MathSciNet)
MR1844385

Zentralblatt MATH identifier
1021.40001

Subjects
Primary: 40A30: Convergence and divergence of series and sequences of functions
Secondary: 40A99: None of the above, but in this section 40C15: Function-theoretic methods (including power series methods and semicontinuous methods)

Keywords
statistical convergence ideals of sets Baire classification of functions

Citation

Kostyrko, Pavel; Wilczyński, Władysław; Šalát, Tibor. I -Convergence. Real Anal. Exchange 26 (2000), no. 2, 669--686. https://projecteuclid.org/euclid.rae/1214571359


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References

  • R. Alexander, Density and multiplicative structure of sets of integers, Acta Arithm., 12 (1967), 321–332.
  • R. G. Bartle, J. I. Joichi, The preservation of convergence of measurable functions under composition, Proc. Amer. Math. Soc., 12 (1968), 122–126.
  • T. C. Brown, A. R. Freedman, The uniform density of sets of integers and Fermat's last Theorem, Compt. Rendus Math. L'Acad. Sci., 12 (1990), 1–6.
  • R. C. Buck, The measure theoretic approach to density , Amer. J. Math., 68 (1946), 560–580.
  • R. C. Buck, Generalized asymptotic density , Amer. J. Math., 75 (1953), 335–346.
  • J. Červe\v nanský, Statistical convergence and statistical continuity , Zbornik Vedeckých Prác M$_t$FSTU, 6 (1998), 207–212.
  • J. S. Connor, The statistical and strong $p$-Cesaro convergence of sequences , Analysis 8 (1988), 47–63.
  • J. S. Connor, Two valued measures and summability, Analysis, 10 (1990), 373–385.
  • J. S. Connor, J. Kline, On statistical limit points and consistency of statistical convergence , J. Math. Appl., 197 (1996), 392–399.
  • H. Fast, Sur la convergence statistique, Coll. Math., 2 (1951), 241–244.
  • J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301–313.
  • J. A. Fridy, Statistical limit points, Proc. Amer. Math. Soc., 118 (1993), 1187–1192.
  • P. Kostyrko, M. Mačaj, T. Šalát and O. Strauch, On statistical limit points, Proc. Amer. Math. Soc., to appear.
  • P. Kostyrko, M. Mačaj, T. Šalát, Statistical convergence and $I$-convergence, Real Analysis Exch., submitted.
  • K. Kuratowski, Topology, Academic Press, Warszawa, 1966.
  • D. Maharam, Finitely additive measures on the integers , Sankhya Indian J. Stat., 38A (1976), 44–59.
  • P. Mikusiński, Axiomatic theory of convergence, (Polish) Uniw. Śląski. Prace Nauk. Prace Matem., 12 (1982), 13–21.
  • H. I. Miller, A measure theoretic subsequence characterization of statistical convergence, Trans. Amer. Math. Soc., 347 (1945), 1811–1819.
  • D. S. Mitrinović, J. Sandor, B. Crstici, Handbook of Number Theory, Kluwer Acad. Publ., Dordrecht-Boston-London, 1996.
  • Jun-iti Nagata, Modern General Topology, North - Holland Publ. Comp., Amsterdam-London, 1974.
  • T. Neubrunn, J. Smital, T, Šalát, On certain properties characterizing locally separable metric spaces, Čas. pešt. mat., 92 (1967), 157–161.
  • H. H. Ostmann, Additive Zahlentheorie I, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1956.
  • G. M. Petersen, Regular Matrix Transformations, Mc Graw-Hill, London-New York-Toronto-Sydney, 1966.
  • B. J. Powell, T. Šalát, Convergence of subseries of the harmonic series and asymptotic densities of sets of positive integers, Publ. Inst. Math. (Beograd), 50 (1964), 60–70.
  • D. Preiss, Approximate derivatives and Baire classes, Czechoslovak Math. J. Z1, 96 (1971), 373–382.
  • T. Šalát, R. Tijdeman, On statistically convergent sequences of real numbers, Math. Slov., 30 (1980), 139–150.
  • T. Šalát, R. Tijdeman, Asymptotic densities of sets of positive integers, Math. Slov., 33 (1983), 199–207.
  • T. Šalát, R. Tijdeman, On density measures of sets of positive integers, Coll. Math. Soc. J. Bolyai 34. Topics in Classical Number Theory, Budapest 1981, 1445–1457.
  • I. J. Schoenberg, The integrability of certain functions and related summability methods , Amer. Math. Monthly, 66 (1959), 361–375.
  • W. Sierpiński, Sur un suite infinie de fonctions continues dont tout fonction d'accumulation est non measurable, Publ. Inst. Math. Beograd, 1 (1947), 5–10