Abstract
In this paper, we introduce the concepts of deferred statistical convergence of order and strongly deferred Cesàro summable functions (real valued) of order on time scales and give some relationships between deferred statistical convergence of order and strongly deferred Cesàro summable functions (real valued) of order on time scales.
1. Introduction
In mathematics, the concept of convergence has been of great importance for many years. This concept has been studied theoretically by many mathematicians in many different fields. Many types of convergence have been defined so far, and then, very valuable results and concepts have been presented to the mathematical community. One of these ideas is the converging statistics. In 1935, Zygmund [1] introduced the concept of statistical convergence to the mathematical community, Steinhaus [2] and Fast [3] independently introduced the concept of statistical convergence, and Schoenberg [4] reintroduced it in the year 1959. Then, it has been addressed under various titles including Fourier analysis, Ergodic theory, Number theory, Turnpike theory, Measure theory, Trigonometric series, and Banach spaces. The concept was later applied to summability theory by various authors such as Çinar et al. ([5, 6]), Çolak [7], Connor [8], Fridy [9], Altay et al. [10], Garcia and Kama [11], Isik et al. ([12–15]), Kucukaslan and Yılmazturk ([16, 17]), Šalát [18], Ercan et al. ([19–21]), and Parida et al. [22], and this concept has been extended to sequence spaces, accordingly, to the notions such as summability theory.
The natural density of a subset of is defined as
provided that limit exists, where is the characteristic function of If
for each , then is said to be statistically convergent to writing . Over the years, that notion has been presented in a variety of ways, and its relationship to aggregation has been investigated in several domains. In recent years, researchers have attempted to apply the relationship between statistical convergence and summability theory in applicable disciplines.
Now is the time to recall the key notions of our study deferred Cesàro mean and deferred statistical convergence.
Deferred Cesàro mean, defined by Agnew [23] in 1932, is a generalization of Cesàro mean, and its definition can be given as follows:
where and are the sequences of nonnegative integers satisfying
Küçükaslan and Yılmaztürk [16, 17] defined the concepts of derferred density and deferred statistical convergence by using the deferred Cesàro mean.
The deferred density of a subset of the natural numbers is defined by
provided the limit exists, where .
A real valued sequence is said to be deferred statistical convergent to , if
Throughout the paper, we assume that the sequences and satisfy the following conditions
and additionally, .
In 1988, Hilger [24] proposed the time scale hypothesis. In 2001, Bohner and Peterson [25] published the first detailed explanation of the time scale theory. In 2003, Guseinov [26] developed a Measure theory on time scales. Cabada and Vivero [27] presented the Lebesque integral on time scales in 2006. These findings provide the foundation for time scale summability theory research. Many mathematicians in various domains have investigated the time scale calculus over the years [28]. As a result, it seems natural to generalize convergence on time scales in light of recent applications of time scales to real-world situations. Numerous writers in the literature have used statistical convergence to apply to time scales for various purposes (see [29–33]).
A nonempty closed subset of real numbers is called a time scale. Two basic concepts on the time scale are the forward jump operator and the backward jump operator. These can be given as follows: (i), (ii) for
A Lebesque -measure is defined on the family of intervals on an arbitrary time scale with the help of forward and backward jump operators. This defined measure is denoted by and provides the following properties: (i)If , then the set is -measurable and (ii)If and , then and (iii)If and , then and
Let be a -measurable subset of , and for , write
Turan and Duman [32, 34] were defined the density and statistical convergence on time scales as follows:
provided that limit exists.
Let be a -measurable function. On , is said to be statistically convergent to if
for every . In this case, we write .
2. Main Results
The aim of this study is to define the concepts of deferred Cesàro summability of order and deferred statistical convergence of order on the time scale and examine the relationships between them.
Definition 1. Let be a -delta measurable function on the time scale and , and we say that is deferred statistically convergent of order on to the number if for every , We will show this convergence with . We will denote by the set of all functions that deferred statistically convergent of order on the time scale
Obviously, (i)If we get and and , then we get the definition of statistical convergence in time scale [32](ii)If we take , , and , then we get lacunary statistical convergence in time scale [34](iii)If we take , , and , then we get statistical convergence in time scale [35](iv)If we get , then we get the definition of deferred statistical convergence in time scale [29]
Example 1. Let be defined as in [36]. It is seen from the following inequality that the function is deferred statistical convergence in for
Theorem 2. If with and , then the following statements hold (i)(ii)
Proof. (i) Let and We write
for every . Taking limit as , (i) will be proved.
(ii) Let . Assume that ; then, the proof of (ii) follows from
Definition 3. Let be a -delta measurable function on the time scale . Then, is strongly deferred Cesàro summable of order to if
By , we denote all strongly deferred Cesàro summable functions of order on .
If we take the function as follows, for ,
which is Cesàro summable.
Theorem 4. Let be a -delta measurable function on the time scale . If , then
Proof. Let and . The proof is obtained from the following inequality:
Corollary 5. Let be a -delta measurable function on the time scale and such that . If , then
To show that the inverse of Theorem 4 and Corollary 5 is not true, we can consider the example on page 3 of Çolak’s article [7].
The converse of Theorem 4 and Corollary 5 is usually not satisfied, but provided that is bounded, by taking , we can give the following result.
Theorem 6. Let be a -delta measurable function on the time scale . If and is bounded, then .
Proof. Suppose that and is bounded. In this case, there is such that and also Therefore, we have It is obvious that for , the theorem is proved.
Theorem 7. Let be a -delta measurable function on the time scale and is bounded. If , then .
Proof. Since , we write Clearly, for , the theorem is proved.
Corollary 8. Let be an arbitrary sequence with , and is bounded. Then, is statistical convergence of order to on implies is deferred statistical convergence of order to on .
Let the four sequences , , , and are nonnegative real numbers such that
for all .
Theorem 9. Let , , , and be given as in (21). If then implies
Proof. Let . The proof is obtained from the following inequality:
Corollary 10. Let , , , and be given as in (21) and such that . If (22) holds, then implies .
Theorem 11. Let , , , and be given as in (21). If then is deferred Cesàro summable to of order on implies is deferred Cesàro summable to of order on .
Proof. Proof follows from the following inequality:
Corollary 12. Let , , , and be given as (21) and . If (24) holds, then is deferred Cesàro summable to of order on implies is deferred Cesàro summable to of order on .
Theorem 13. Let , , , and be given as (21). If If and is bounded, then
Proof. Suppose that . Since is bounded, there exists a positive number such that . Then, we may write This completes the proof.
3. Conclusion
Various variations of statistical convergence have been studied throughout the years, yielding some extremely important conclusions. Deferred statistical convergence of order is one of these versions. This variant of statistical convergence is investigated on arbitrary time scales in this paper, and a significant generalization is made. As a result, the current results constitute a particular case of our findings. Then, on temporal scales, strongly postponed Cesàro summability of order is built. Finally, various inclusion relations for the newly obtained spaces are investigated. The concepts and theorems mentioned will vary as the time scale changes. This will have a significant impact on applications employing the notion of summability in numerous ways.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally, and they have read and approved the final manuscript for publication.