\(\Delta\)-Convergence and Uniform Distribution in Lacunary Sense

Authors

  • B. Aris Department of Mathematic, Mersin University, Mersin
  • Mehmet Küçükaslan Department of Mathematic, Mersin University, Mersin

DOI:

https://doi.org/10.26713/cma.v8i1.578

Keywords:

Convergence of sequence, Statistically convergence, Uniformly distribution of sequence, Lacunary convergence

Abstract

In this paper, by considering usual partition of \([0, \infty)\) \(\Delta\)-convergence of non-negative real valued sequences is defined. It is shown that every convergent sequence is \(\Delta\)-convergence but the converse is not true, in general. Besides, some basic properties of \(\Delta\)-convergence as well as the second part of this paper by using any lacunary sequences as a partition of non-negative real numbers, lacunary uniform distribution is defined and some inclusion result between uniform distribution modulo 1 and lacunary uniform distribution has been given.

Downloads

Download data is not yet available.

References

R.E. Burkard, Thesis, Univ. of Vienna (1967).

R.E. Burkard, Sitzgsber. í–sterr. Akad. Wiss. Math. Naturw. K1., Abt. II, 177 (1969), 395–406.

J. Cigler, Monatsh. Math. 64 (1960), 201–225.

J. Cigler and G. Helmberg, Jber. Deutsch. Math.-Verein. 64 (1961), 1–50.

J. Connor, Jour. of Math. Anal. and Appl. 197 (1996), 392–399.

H. Davenport, Mathematika 3, (1956).

H. Davenport and P. Erdös, Canad. J. Math. (1952), 58–63.

H. Davenport and W.J. Le Veque, Michigan Math. J. 10 (1963), 315–319.

A.R. Freedman, J. Sember and M. Raphael Proc. London Math. Soc. 37 (3) (1978), 508–520.

J.A. Fridy, Analysis 5 (1985), 301–313.

J.F. Koksma, Erg. Math. Grenzgeb. Band 4, Heft 4, Julius Springer, Berlin (1936).

L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley & Sons, New York (1974).

W.J. Le Veque, Pasific J. Math. 3 (1953), 757–771.

W. M. Schmidt, Studia Sci. Math. Soc. 25 (1970), 608–614.

H. Weyl, Nachr. Ges. Wiss. Göttingen, Math.-Phys. (1914), 234–244.

H. Weyl, Math. Ann. 77 (1916), 313–352.

Downloads

Published

30-06-2017
CITATION

How to Cite

Aris, B., & Küçükaslan, M. (2017). \(\Delta\)-Convergence and Uniform Distribution in Lacunary Sense. Communications in Mathematics and Applications, 8(1), 69–86. https://doi.org/10.26713/cma.v8i1.578

Issue

Vol. 8 No. 1 (2017)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.