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Uniform Statistical Convergence on Time Scales
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.
The idea of statistical convergence was known to Zygmund  as early as 1935 and in particular after 1951 when Fast  and Steinhaus  reintroduced statistical convergence for sequences of real numbers. Later, Schoenberg  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 ).
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 ).
A sequence is said to be -summable to a number if as -summability reduces to summability when (see ).
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  as follows.
Let and define the -density of by reduces to the asymptotic density in case of for all (see ).
A sequence is said to be -statistically convergent to if for every (see )
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 ). 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 ).
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 ). 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 , Brown and Freedman , and Maddox .
The notion of -uniform statistical convergence is first introduced by Nuray  as follows.
Definition 2 (see ). 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 , Borwein introduced and studied strongly summable functions. His definition is as follows.
Definition 5 (see ). 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  studied -strong summable and -statistically convergent functions as in the following.
Definition 6 (see ). 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 , Nuray and Aydin , and Et et al. .
There are some studies about statistical convergence on time scales in the literature. For instance, Seyyidoglu and Tan  gave some new notations such as -convergence and -Cauchy by using -density and investigated their relations. Turan and Duman  introduced the concept of density and statistical convergence of delta measurable real-valued functions defined on time scales as follows.
Definition 7 (see ). 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 .
Definition 8 (see ). 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 .
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  as follows.
Definition 17 (see ). 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  and Lebesgue -integral on time scales introduced by Cabada and Vivero .
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 .
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]).
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.
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