# Examination of Generalized Statistical Convergence of Orderα on Time Scales

Authors: Muhammed Çinar; Mahmut Işik
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we introduce the concepts of deferred statistical convergence of orderα and strongly deferred Cesàro summable functions (real valued) of order α on time scales and give some relationships between deferred statistical convergence of order α and strongly deferred Cesàro summable functions (real valued) of order α on time scales.


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## Body

## 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 subsetA of ℕ is defined as
(1)δA=limn⟶∞1n∑k=1nχAk,provided that limit exists, whereχA is the characteristic function of A. If
(2)δk∈ℕ:xk−L≥ε=0,for eachε>0, then x=xk is said to be statistically convergent to ℓ writing S−limk⟶∞xk=ℓ. 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:
(3)Dp,qxm=1qm−pm∑pm+1qmxk,wherep=pm and q=qm are the sequences of nonnegative integers satisfying
(4)pm<qmandlimm⟶∞qm=∞.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 subsetA of the natural numbers ℕ is defined by
(5)δp,qA=limm⟶∞1qm−pmAp,qm,provided the limit exists, whereAp,qm=pm<k≤qm:k∈A.A real valued sequencex=xk is said to be deferred statistical convergent to L, if
(6)limm⟶∞1qm−pmpm<k≤qm:xk−L≥ε=0,for eachε>0 [16, 17].Throughout the paper, we assume that the sequencespm and qm satisfy the following conditions
(7)pm<qmandlimm⟶∞qm=∞,and additionally,limm⟶∞qm−pm=∞.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)
σ:T⟶T, σs=inft∈T:t>s(ii)
ρs=supt∈T:t<s for s∈TA LebesqueΔ-measure is defined on the family of intervals x,yT=s∈T:x≤s≤y on an arbitrary time scale T with the help of forward and backward jump operators. This defined measure is denoted by νΔ and provides the following properties:
(i)
Ifx∈T\maxT, then the set x is Δ-measurable and νΔx=σx−x(ii)
Ifx,y∈T and x≤y, then νΔx,yT=y−x and νΔx,yT=y−σx(iii)
Ifx,y∈T\maxT and x≤y, then νΔx,yT=σy−σx and νΔx,yT=σy−xLetΩ be a Δ-measurable subset of T, and for t∈T, write
(8)Ωt=s∈t0,t:s∈Ω.Turan and Duman [32, 34] were defined the density and statistical convergence on time scales as follows:
(9)δTΩ=limt⟶∞νΔΩtνΔt0,t,provided that limit exists.Letf:T⟶R be a Δ-measurable function. On T, f is said to be statistically convergent to L if
(10)δTt∈T:ft−L≥ε=0,for everyε>0. In this case, we write stT−limt⟶∞ft=L.


## 2. Main Results

The aim of this study is to define the concepts of deferred Cesàro summability of orderα and deferred statistical convergence of order α on the time scale and examine the relationships between them.Definition 1.
Letf be a Δ-delta measurable function on the time scale T and α∈0,1, and we say that f is deferred statistically convergent of order α on T to the number L if for every ε>0,
(11)limm⟶∞νΔk∈pm,qmT:fk−L⩾ενΔαpm,qmT=0.
We will show this convergence withDstTα−limft=L. We will denote by DSTαp,q the set of all functions that deferred statistically convergent of order α on the time scale T.Obviously,(i)
If we getqs=s and ps=s0 and α=1, then we get the definition of statistical convergence in time scale [32](ii)
If we takeqm=km, pm=km−1, and α=1, then we get lacunary statistical convergence in time scale [34](iii)
If we takeqt=t, pt=t−λt+t0, and α=1, then we get λ statistical convergence in time scale [35](iv)
If we getα=1, then we get the definition of deferred statistical convergence in time scale [29]Example 1.
Letf be defined as in [36]. It is seen from the following inequality that the function f is deferred statistical convergence in T for α>1/2(12)νΔk∈pm,qmT:fk≥ενΔαpm,qmT≤νΔpm,σpm+vmvmα=vmvmα⟶0m⟶∞.Theorem 2.
Iff,g:T⟶ℝ with DstTα−limft=L1 and DstTα−limgt=L2, then the following statements hold
(i)
DstTα−limft+gt=L1+L2(ii)
DstTα−limcft=cL1Proof.
(i) LetDstTα−limft=L1 and DstTα−limgt=L2. We write
(13)νΔk∈pm,qmT:fk+gk−L1+L2⩾ενΔαpm,qmT=νΔk∈pm,qmT:fk−L1⩾ενΔαpm,qmT+νΔk∈pm,qmT:gk−L2⩾ενΔαpm,qmT,
for everyε>0. Taking limit as m⟶∞, (i) will be proved.
(ii) LetDstTα−limft=L. Assume that c≠0; then, the proof of (ii) follows from
(14)νΔk∈pm,qmT:cfk−cL⩾ενΔαpm,qmT=νΔk∈pm,qmT:fk−L⩾ε/cνΔαpm,qmT.Definition 3.
Letf be a Δ-delta measurable function on the time scale T. Then, f is strongly deferred Cesàro summable of order α to L if
(15)limm⟶∞1νΔαpm,qmT∫pm,qmTfk−LΔk=0.
ByDTαp,q, we denote all strongly deferred Cesàro summable functions of order α on T.
If we take the functionf as follows, for α>1/2,
(16)ft=1,ifs∈pm,σpm+1,T1,ifs∈σpm+1,σpm+2T,⋯1,ifs∈σpm+vm−1,σpm+vmT,0,otherwise,
which is Cesàro summable.Theorem 4.
Letf be a Δ-delta measurable function on the time scale T. If f∈DTαp,q, then f∈DSTαp,q.Proof.
Letε>0 and Dε=s∈pm,qmT:fs−L⩾ε. The proof is obtained from the following inequality:
(17)∫pm,qmTfs−LΔs⩾∫Dεfs−LΔs⩾εμΔDε.Corollary 5.
Letf be a Δ-delta measurable function on the time scale T and α,β∈0,1 such that α<β. If f∈DTαp,q, then f∈DSTβp,q.
To show that the inverse of Theorem4 and Corollary 5 is not true, we can consider the example on page 3 of Çolak’s article [7].
The converse of Theorem4 and Corollary 5 is usually not satisfied, but provided that f is bounded, by taking α=1, we can give the following result.Theorem 6.
Letf be a Δ-delta measurable function on the time scale T. If f∈DSTp,q and f is bounded, then f∈DTp,q.Proof.
Suppose thatf∈DSTp,q and f is bounded. In this case, there is K>0 such that fk⩽K and also
(18)limm⟶∞1νΔpm,qmνΔk∈pm,qm:fk−L⩾ε=0.
Therefore, we have(19)1νΔpm,qm∫pm,qmTfk−LΔk=1νΔDpm,qm∫Dεfk−LΔk+1νΔpm,qm∫pm,qmT\Dεfk−LΔk⩽KνΔpm,qm∫DεΔk+ενΔpm,qm∫pm,qmTΔk=KνΔDενΔpm,qm+ε.
It is obvious that form⟶∞, the theorem is proved.Theorem 7.
Letf be a Δ-delta measurable function on the time scale T and νΔαpm,qm/σqmα is bounded. If stTα−limft=L, then DstTα−limft=L.Proof.
SincestTα−limft=L, we write
(20)1νΔαt0,qmμΔk∈t0,qmT:fk−L⩾ε⩾νΔDεσqm−t0α⩾σqm−σpmασqmανΔDεσqm−σpmα.
Clearly, form⟶∞, the theorem is proved.Corollary 8.
Letqm be an arbitrary sequence with qm∈t0,t, and νΔαt0,t/νΔαpm,qm is bounded. Then, f is statistical convergence of order α to L on T implies f is deferred statistical convergence of order α to L on T.
Let the four sequencespm, qm, p′m, and q′m are nonnegative real numbers such that
(21)pm⩽p′m<q′m⩽qm,
for allm∈ℕ.Theorem 9.
Letpm, qm, p′m, and q′m be given as in (21). If
(22)limm⟶∞νΔαp′m,q′mνΔαpm,qm>0,
thenf∈DSTαp,q implies f∈DSTαp′,q′.Proof.
LetD′ε=k∈p′m,q′m:fk−L⩾ε. The proof is obtained from the following inequality:
(23)1νΔαpm,qmνΔDε⩾νΔαp′m,q′mνΔαpm,qm1νΔαp′m,q′mνΔD′ε..Corollary 10.
Letpm, qm, p′m, and q′m be given as in (21) and α,β∈0,1 such that α≤β. If (22) holds, then f∈DSTαp,q implies f∈DSTβp′,q′.Theorem 11.
Letpm, qm, p′m, and q′m be given as in (21). If
(24)limm⟶∞νΔαpm,qmνΔαp′m,q′m>0,
thenf is deferred Cesàro summable to L of order α on pm,qm implies f is deferred Cesàro summable to L of order α on p′m,q′m.Proof.
Proof follows from the following inequality:(25)1νΔαpm,qm∫pm,qmTfk−LΔk⩾νΔαp′m,q′mνΔαpm,qm1νΔαp′m,q′m∫p′m,q′mTfk−LΔk.Corollary 12.
Letpm, qm, p′m, and q′m be given as (21) and α,β∈0,1,α≤β. If (24) holds, then f is deferred Cesàro summable to L of order α on pm,qm implies f is deferred Cesàro summable to L of order α on p′m,q′m.Theorem 13.
Letpm, qm, p′m, and q′m be given as (21). If
(26)limm⟶∞νΔpm,p′mνΔαp′m,q′m>0,limm⟶∞νΔqm,q′mνΔαp′m,q′m>0.
Iff∈DSTαp′,q′ and f is bounded, then f∈DTαp,q.Proof.
Suppose thatf⟶LDSTαp′,q′. Since f is bounded, there exists a positive number K such that fs⩽K. Then, we may write
(27)1νΔαpm,qm∫pm,qmTfk−LΔk=1νΔαpm,qm∫pm,p′mTfk−LΔk+∫p′m,q′mTfk−LΔk+∫q′m,qmTfk−LΔs⩽1νΔαpm,qm∫pm,p′mTKΔk+∫p′m,q′mTfk−LΔk+K∫q′m,qmTΔk⩽1νΔαpm,qmσp′m−σpm+σqm−σq′m+1νΔαpm,qm∫pm,qmT:fk−L⩾εfk−LΔk+∫pm,qmT:fk−L<εfk−LΔk⩽μΔpm,p′m+νΔqm,q′mνΔαp′m,q′m.K+KνΔαp′m,q′mνΔp′m<k1⩽q′m:fk−L⩾ε+ενΔαp′m,q′m.
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 orderα 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 α 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.



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