Journal of Applied Mathematics

Volume 2014 (2014), Article ID 471437, 6 pages

http://dx.doi.org/10.1155/2014/471437

Research Article

Uniform Statistical Convergence on Time Scales

Yavuz Altin, Hikmet Koyunbakan, and Emrah Yilmaz

Department of Mathematics, Firat University, 23119 Elazıg, Turkey

Received 22 March 2014; Accepted 30 May 2014; Published 18 June 2014

Academic Editor: Alberto Cabada

Copyright © 2014 Yavuz Altin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We will introduce the concept of - and -uniform density of a set and - and -uniform statistical convergence on an arbitrary time scale. However, we will define -uniform Cauchy function on a time scale. Furthermore, some relations about these new notions are also obtained.

1. Introduction

The idea of statistical convergence was known to Zygmund [1] as early as 1935 and in particular after 1951 when Fast [2] and Steinhaus [3] reintroduced statistical convergence for sequences of real numbers. Later, Schoenberg [4] independently gave some basic properties of statistical convergence. Several generalizations and applications of this notion have been discussed in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory, and Banach spaces under different names (see [5–11]).

Statistical convergence depends on the density of subsets of the set . Recall that a subset of is said to have “asymptotic density” if

where the vertical bars denote the cardinality of the enclosed set. It is clear that any finite subset of has zero asymptotic density and (see [12]).

A sequence is said to be statistically convergent to a real number if

for each . And we write or . The set of all statistically convergent sequences is denoted by (see [2, 3, 5, 6, 9, 13]).

The generalized de la Vallée-Poussin mean is defined by

where is a nondecreasing sequence of positive numbers such that as and . The set of all such sequences will be denoted by (see [14]).

A sequence is said to be -summable to a number if as -summability reduces to summability when (see [14]).

We write

for the sets of sequences which are strongly Cesàro summable and strongly -summable, respectively. Strong -summability reduces to strong summability when .

The notion of -statistical convergence was introduced by Mursaleen [15] as follows.

Let and define the -density of by

reduces to the asymptotic density in case of for all (see [15]).

A sequence is said to be -statistically convergent to if for every (see [15])

After the concept of almost -statistical convergence was studied by Savaş [16], many authors have studied statistical convergence (see [6, 9, 10]).

Statistical convergence is applied to time scales for different purposes by various authors (see [17, 18]).

We here recall some basic concepts and notations from the theory of time scales. A time scale is an arbitrary nonempty closed set of real numbers. We use the symbol to denote a time scale. A time scale has the topology that it inherits from the real numbers with the standard topology. The theory of time scale was introduced by Hilger in his Ph.D. thesis supervised by Auldbach in 1988 (see [19]). It allows unifying the usual differential and integral calculus for one variable. One can replace the range of definition of the functions under consideration by an arbitrary time scale . Now, time scale theory has been applied to different areas by many authors (see [20–24]).

The forward jump operator can be defined by

for . And the graininess function is defined by . In this definition, we put , where is an empty set. A half open interval on an arbitrary time scale is given by

Open intervals or closed intervals can be defined similarly (see [20, 21]).

Now, let denote the family of all left closed and right open intervals of of the form . Let be the set function on such that

Then, it is known that is a countably additive measure on . Now, the Caratheodory extension of the set function associated with family is said to be the Lebesgue -measure on and is denoted by . In this case, it is known that if , then the single point set is -measurable and . If and , then . If , and (see [18]).

In this study, we will give some notations for -uniform and -uniform density of a set and -uniform and -uniform statistical convergence and some properties of -uniform and -uniform statistical convergence on time scales.

Definition 1 (see [25]). A subset of is said to be uniformly dense if

uniformly in or, equivalently,

uniformly in , where and is characteristic function. Subsequently, uniform density was studied by Baláž and Šalát [26], Brown and Freedman [27], and Maddox [28].

The notion of -uniform statistical convergence is first introduced by Nuray [29] as follows.

Definition 2 (see [29]). Let be a real or complex valued sequence. If

uniformly in is said to be -uniform statistically convergence to for .

Based on this notion, we give the following definitions to generalize -uniform statistical convergence.

Definition 3. Let and define the -uniform density of by

reduces to the in case of for all .

Definition 4. A sequence is said to be -uniform statistically convergent to if

for every uniformly in .

In [30], Borwein introduced and studied strongly summable functions. His definition is as follows.

Definition 5 (see [30]). A real-valued function , measurable (in Lebesgue sense) on the interval , is said to be strongly summable to if

will denote the space of real-valued function , measurable (in the Lebesgue sense) on the interval .

Furthermore, Nuray [31] studied -strong summable and -statistically convergent functions as in the following.

Definition 6 (see [31]). Let , let be a real number, and let be a real-valued function which is measurable (in Lebesgue sense) on the interval , if

then, one says that is strongly summable to . Strongly summable number sequences and statistically convergent number sequences were studied by Maddox [32], Nuray and Aydin [33], and Et et al. [34].

There are some studies about statistical convergence on time scales in the literature. For instance, Seyyidoglu and Tan [17] gave some new notations such as -convergence and -Cauchy by using -density and investigated their relations. Turan and Duman [18] introduced the concept of density and statistical convergence of delta measurable real-valued functions defined on time scales as follows.

Definition 7 (see [18]). Suppose that is a -measurable subset of . Then, for , one defines the set by

In this case, one defines the density of on , denoted by ,

provided that the above limit exists. Furthermore, is statistically convergent to a real number on if, for every ,

where is a -measurable function (see [17, 18]). Lebesgue -measure is introduced by Guseinov [20].

Definition 8 (see [18]). Let be a -measurable function. is statistical Cauchy on if, for each , there exists a number such that

2. Main Results and Preliminaries

It is well known that the notion of statistical convergence is closely related to the density of the subset of . So, in this section, we will first define -uniform and -uniform density of the subset of the time scale. By using these definitions, we will focus on constructing a concept of -uniform (or -uniform) statistical convergence and -uniform statistical Cauchy function on time scales. In following definitions, notations and shows that depends on and , respectively.

Definition 9. Let be a -measurable subset of . Then, one defines the set by

for . In this case, one defines the -uniform density of on , denoted by , as follows:

provided that the above limit exists.

Definition 10. Let be a -measurable function. Then, one says that is -uniform statistically convergent to a real number on if

uniformly in for every . In this case, one writes . The set of all -uniform statistically convergent functions on will be denoted by .

In case of is for and . In this instance, -uniform statistical convergence on time scales is reduced to classical -uniform statistical convergence which is given by Definition 2. This shows that our results are generalizations of classical results.

Similarly, we can define -uniform statistical Cauchy functions on a time scale based on Definition 8.

Definition 11. Let be a -measurable function. is an -uniform statistical Cauchy function on if there exists a number such that

for each uniformly in . One can easily see that this definition is a generalization of Definition 8.

Definition 12. Let be a -measurable subset of . Then, one defines the set by

for . In this case, one defines the -uniform density of on denoted by , as follows:

provided that the above limit exists.

Definition 13. Let be a -measurable function. One says that is -uniform statistically convergent to a real number on if

uniformly in for every . In this case, one writes . The set of all -uniform statistically convergent functions on will be denoted by .

Hence, we have generalized Definition 3 to an arbitrary time scale. We can easily get classical -uniform statistical convergence by taking in Definition 13.

Proposition 14. If with and , then the following statements hold:

(i), (ii) .

Theorem 15. For to be any -measurable function, is -uniform statistically convergent on if and only if is a -uniform statistical Cauchy function on .

Proof. We can prove this by using techniques similar to Theorem 3 of [29].

Theorem 16. Consider if and only if

Proof. For given , we have

Therefore,

Hence by using (28) and taking the limit as , we get which implies .

The definition of -Cesàro summability on time scales was given by Turan and Duman [18] as follows.

Definition 17 (see [18]). Let be a -measurable function and . Then, is strongly -Cesàro summable on if there exists some such that

The set of all -Cesàro summable functions on will be denoted by .

Measure theory on time scales was first constructed by Guseinov [20] and Lebesgue -integral on time scales introduced by Cabada and Vivero [35].

Definition 18. Let be a -measurable function and . One says that is uniformly strongly -summable on if there exists some such that

In this case, one writes . The set of all uniformly strongly -summable functions on will be denoted by .

Lemma 19. Let be a -measurable function and

for . In this case, we have

Proof. This can be proved by using a method similar to the approach in [18].

Theorem 20. Let be a -measurable function, , and . Then, one gets the following.

(i).(ii)If is uniformly strongly -summable to , then .(iii)If   and is a bounded function, then is uniformly strongly -summable to .

Proof. (i) Let and . We can write

Therefore, implies .

(ii) Let be uniformly strongly -summable to . For given , let

on time scale . Then, it follows from Lemma 19 that

Dividing both sides of the last inequality by and taking limit as , we obtain

which yields .

(iii) Let be bounded and statistically convergent to on . Then, there exists a positive number such that for all , and also

where is as before. Since

we obtain

Since is arbitrary, the proof follows from (39) and (41).

Theorem 21. Let be a measurable function. Then, if and only if there exists a measurable set such that and .

Proof. It can be easily proved by using similar way in the study of Turan and Duman (see [18, Theorem 3.9]).

3. Conclusions

In this study, we introduced the historical development of the notion of statistical convergence. Then we presented some fundamental notions based on statistical convergence. The concepts of - and -uniform density and uniform statistical convergence were defined on an arbitrary time scale. However, we defined -uniform Cauchy functions on a time scale in general. Furthermore, we obtained some relations between these new notions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, UK, 1979.

H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951.

H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloquium Mathematicum, vol. 2, pp. 73–74, 1951.

I. J. Schoenberg, “The integrability of certain functions and related summability methods.,” The American Mathematical Monthly, vol. 66, pp. 361–375, 1959.

J. S. Connor, “The statistical and strong -Cesàro convergence of sequences,” Analysis, vol. 8, no. 1-2, pp. 47–63, 1988.

J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985.

I. J. Maddox, “Statistical convergence in a locally convex space,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104, no. 1, pp. 141–145, 1988.

D. Rath and B. C. Tripathy, “On statistically convergent and statistically Cauchy sequences,” Indian Journal of Pure and Applied Mathematics, vol. 25, no. 4, pp. 381–386, 1994.

T. Šalát, “On statistically convergent sequences of real numbers,” Mathematica Slovaca, vol. 30, no. 2, pp. 139–150, 1980.

B. C. Tripathy, “On statistical convergence,” Proceedings of the Estonian Academy of Sciences Physics, vol. 47, no. 4, pp. 299–303, 1998.

F. Móricz, “Statistical limits of measurable functions,” Analysis, vol. 24, no. 1, pp. 1–18, 2004.

R. C. Buck, “Generalized asymptotic density,” American Journal of Mathematics, vol. 75, pp. 335–346, 1953.

C. Belen and S. A. Mohiuddine, “Generalized weighted statistical convergence and application,” Applied Mathematics and Computation, vol. 219, no. 18, pp. 9821–9826, 2013.

L. Leindler, “Über die verallgemeinerte de la Vallée-Poussinsche Summierbarkeit allgemeiner Orthogonalreihen,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 16, pp. 375–387, 1965.

Mursaleen, “-statistical convergence,” Mathematica Slovaca, vol. 50, no. 1, pp. 111–115, 2000.

E. Savaş, “Strong almost convergence and almost -statistical convergence,” Hokkaido Mathematical Journal, vol. 29, no. 3, pp. 531–536, 2000.

M. Seyyit Seyyidoglu and N. Ö. Tan, “A note on statistical convergence on time scale,” Journal of Inequalities and Applications, vol. 2012, article 219, 2012.

C. Turan and O. Duman, “Statistical convergence on timescales and its characterizations,” in Advances in Applied Mathematics and Approximation Theory, vol. 41 of Springer Proceedings in Mathematics and Statistics, pp. 57–71, Springer, New York, NY, USA, 2013.

S. Hilger, “Analysis on measure chains–-a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.

G. Sh. Guseinov, “Integration on time scales,” Journal of Mathematical Analysis and Applications, vol. 285, no. 1, pp. 107–127, 2003.

M. Bohner and A. Peterson, Dynamic Equations on Time Scales, an Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001.

M. Bohner and G. Sh. Guseinov, “Multiple integration on time scales,” Dynamic Systems and Applications, vol. 14, no. 3-4, pp. 579–606, 2005.

M. Bohner and G. Sh. Guseinov, “Multiple Lebesgue integration on time scales,” Advances in Difference Equations, vol. 2006, Article ID 026391, 2006.

R. Agarwal, M. Bohner, D. O'Regan, and A. Peterson, “Dynamic equations on time scales: a survey,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 1–26, 2002.

R. A. Raimi, “Convergence, density, and -density of bounded sequences,” Proceedings of the American Mathematical Society, vol. 14, pp. 708–712, 1963.

V. Baláž and T. Šalát, “Uniform density and corresponding -convergence,” Mathematical Communications, vol. 11, no. 1, pp. 1–7, 2006.

T. C. Brown and A. R. Freedman, “The uniform density of sets of integers and Fermat's last theorem,” La Société Royale du Canada. L'Académie des Sciences, vol. 12, no. 1, pp. 1–6, 1990.

I. J. Maddox, “A new type of convergence,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 83, no. 1, pp. 61–64, 1978.

F. Nuray, “Uniform statistical convergence,” Science and Engineering Journal of Firat University, vol. 11, no. 3, pp. 219–222, 1999.

D. Borwein, “Linear functionals connected with strong Cesàro summability,” Journal of the London Mathematical Society, vol. 40, pp. 628–634, 1965.

F. Nuray, “-strongly summable and -statistically convergent functions,” Iranian Journal of Science and Technology, Transaction A: Science, vol. 34, no. 4, pp. 335–338, 2010.

I. J. Maddox, “Spaces of strongly summable sequences,” The Quarterly Journal of Mathematics, vol. 18, no. 2, pp. 345–355, 1967.

F. Nuray and B. Aydin, “Strongly summable and statistically convergent functions,” Informacines Technologijos Ir Valdymas, vol. 1, no. 30, pp. 74–76, 2004.

M. Et, S. A. Mohiuddine, and A. Alotaibi, “On -statistical convergence and strongly -summable functions of order ,” Journal of Inequalities and Applications, vol. 2013, article 469, 2013.

A. Cabada and D. R. Vivero, “Expression of the Lebesgue -integral on time scales as a usual Lebesgue integral: application to the calculus of -antiderivatives,” Mathematical and Computer Modelling, vol. 43, no. 1-2, pp. 194–207, 2006.