Convergence of Multiple Series and ℐ–Variation

Baláž, Vladimír; Faisant, Alain; Grekos, Georges; Kostra, Juraj

doi:10.2478/udt-2025-0008

Abstract

In the classical theory of multiple series the notion of variation of a sequence plays an important role. According to the concept of ℐ−convergence the notion of ℐ−variation of a sequence was recently introduced. In this paper we will examine applications of ℐ−variation of a sequence mainly in the theory of multiple series in particular in relation with the notion of ℐ−convergence.

References

BALÁŽ, V.—VISNIAY, T.: ℐ−convergence of arithmetical functions. In: Number Theory and Its Applications, chapter 8, IntechOpen, London, 2020, pp. 125–145,
Search in Google Scholar Back to article
BALÁŽ, V.—FAISANT, A.—GREKOS, G.: On Real Algebras Associated with Ideal Convergence, Real Anal. Exchange 47 (2022), 281–296.
Search in Google Scholar Back to article
BALCERZAK, M.—DEMS, K.—KOMISARSKI, A.: Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007), no. 1, 715–729.
Search in Google Scholar Back to article
BARTOSZEWICZ, A.—G LA¸B, S.—WACHOWICZ, A.: Remarks on ideal boundedness, convergence and variation of sequences, J. Math. Anal. Appl. 375 (2011), 431–435.
Search in Google Scholar Back to article
FAISANT, A.—GREKOS, G.—TOMA, V.: On the statistical variation of sequences, J. Math. Anal. Appl. 306 (2005), 432–439.
Search in Google Scholar Back to article
FAST, H.: Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.
Search in Google Scholar Back to article
FURSTENBERG, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1981.
Search in Google Scholar Back to article
HARDY, G. H.: On convergence of certain multiple series, Proc. Cambridge Phil. Soc. 19 (1917), 86–95.
Search in Google Scholar Back to article
JIN, R.: Applications of nonstandard analysis in additive number theory, Bull. Symbolic Logic 6 (2000), no. 3, 331–341.
Search in Google Scholar Back to article
KOSTYRKO, P.—ŠALÁT, T.—WILCZYŃSKI, W.: I-convergence, Real Anal. Exchange 26 (2000/2001), no. 2, 669–686.
Search in Google Scholar Back to article
PAŠTÉKA, M.: Density and Related Topics. VEDA Bratislava, AC Praha, 2017.
Search in Google Scholar Back to article
SCHOENBERG, I. J.: The integrability of certain functions and related summability methods, Amer. Math. Monthly 66, 1959, 361–375.
Search in Google Scholar Back to article
STRAUCH, O.—Š. PORUBSKÝ, V.: Distribution of Sequences: A Sampler. Schr. Slowak. Akad. Wiss. Vol. 1 [Series of the Slovak Academy of Sciences] Peter Lang, Frankfurt am Main, 2005.
Search in Google Scholar Back to article
ŠALÁT, T.: Infinite Series. Academia, Praha, 1974. (In Slovak)
Search in Google Scholar Back to article

Convergence of Multiple Series and ℐ–Variation

Abstract

Paradigm

My account