International Journal of Analysis and Applications
ISSN 2291-8639
Volume 14, Number 1 (2017), 99-106
http://www.etamaths.com

UNIFORM LACUNARY STATISTICAL CONVERGENCE ON TIME SCALES

E. YILMAZ1, S. A. MOHIUDDINE2,∗, Y. ALTIN1 AND H. KOYUNBAKAN1

Abstract. We introduce (θ,m)-uniform lacunary density of any set and (θ,m)-uniform lacunary

statistical convergence on an arbitrary time scale. Moreover, (θ,m)-uniform strongly p-lacunary

summability and some inclusion relations about these new concepts are also presented.

1. Introduction and preliminaries

The idea of statistical convergence goes back to the study of Zygmund [42] which was published in
1935. Statistical convergence of number sequences was formally introduced by Fast [13] and Steinhaus
[40] independently in the same year. Over the years and under different names, statistical convergence
has been discussed in the theory of Fourier analysis, ergodic theory, number theory, approximation
theory, measure theory, trigonometric series, turnpike theory and Banach spaces. Later on, it was
further investigated from the sequence space point of view and linked with summability theory by
Fridy [15], Connor [8], Maddox [23], Rath and Tripathy [33], Tripathy [37], Moricz [28], Belen and
Mohiuddine [3], Braha et al. [5], Edely et al. [10], Mohiuddine et al . [26] and references therein.

The statistical convergence is related to the density of subsets of N. The asymptotic density of a
set A ⊂ N is defined by

δ (A) = lim
n→∞

1

n
|{k ≤ n : k ∈ A}| ,

whenever the limit exists. Here, |{k ≤ n : k ∈ A}| indicates the number of elements of A ⊆ N not
exceeding n. Any finite subset of N has zero asymptotic density and δ (Ac) = 1 − δ (A). A sequence
(xk) is statistically convergent [13] to a real number L if for each ε > 0,

δ ({k ∈ N: |xk −L| ≥ ε}) = 0.

In this case, S- lim xk = L. The set of all statistically convergent sequences is denoted by S.
By a lacunary sequence θ = (kr) (r = 0, 1, 2, ...), where k0 = 0, we shall mean an increasing sequence

of non- negative integers with kr−kr−1 →∞ as r →∞. The intervals determined by θ will be denoted
by Ir = (kr−1,kr] and hr = kr −kr−1 where qr =

kr
kr−1

(see [14]). The space of all lacunary strongly

convergent sequences Nθ was defined by Freedman et al. as follows

Nθ =

{
x = (xk) : lim

r→∞

(
1

hr

∑
k∈Ir

|xk −L|

)
= 0, for some L

}
.

To understand lacunary sequences, we need to consider below examples.

Example 1.1. θ =
(
r2
)

is a lacunary sequence. Let us check the above conditions. One can easily see

that 0 < r2 < (r + 1)2. So, θ is an increasing sequence where k0 = 0. Furthermore, hr →∞ as r goes
to infinite as shown in following table:

hr = kr −kr−1 h1 h2 h3 h4 ... h10 h11 ... h100 ... hr
kr = r

2 1 3 5 7 → 19 21 → 199 → 2r − 1

Table 1. hr →∞ as r goes to infinite.

Received 16th February, 2017; accepted 5th April, 2017; published 2nd May, 2017.

2010 Mathematics Subject Classification. 40A35, 46A45, 34N05.
Key words and phrases. uniform lacunary statistical convergence; sequence spaces; time scale.

c©2017 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

99



100 YILMAZ, MOHIUDDINE, ALTIN AND KOYUNBAKAN

Example 1.2. θ = (r) is not a lacunary sequence. Although the first two conditions satisfy, hr does
not go to infinite as r →∞ as seen in following table:

hr = kr −kr−1 h1 h2 h3 h4 ... h10 h11 ... h100 ... hr
kr = r 1 1 1 1 → 1 1 → 1 → 1

Table 2. hr → 1 as r goes to infinite.
Let K ⊂ N. One defines the θ-density [16] of a set K by

δθ (K) = lim
r→∞

1

hr
|K ∩ Ir| .

By using lacunary sequences, Fridy and Orhan [16] studied a related concept of convergence in which
{k : k ≤ n} is replaced by {k : kr−1 < k ≤ kr} for a lacunary sequence θ = {kr} as follows: A real or
complex sequence (xk) is lacunary statistically convergent to L if for every ε > 0,

lim
r→∞

1

hr
|{k ∈ Ir : |xk −L| ≥ ε}| = 0.

In this case, Sθ- lim x = L. Lacunary statistical convergence and related notions were studied by many
authors (see [7], [9], [11], [12], [17], [20], [24], [25], [27], [29]). Furthermore, Nuray and Aydın [31]
introduced and studied strongly lacunary summable functions. Here, our aim is to move some notions
and properties about lacunary sequence to time scale calculus. Before our new concepts, we recall the
main features of the time scale theory.

A time scale T is an arbitrary, nonempty, closed subset of real numbers. The calculus of time scale
was introduced by Hilger in his Ph.D. thesis supervised by Auldbach in 1988 (see [21], [22]). It allows
to unify the usual differential and integral calculus for one variable. One can replace the range of
definition R of the functions under consideration by an arbitrary time scale T. Recently, time scale
theory has been applied to different areas by many authors (see [4], [18], [19]). The followings notions
are very important for this theory.

Forward jump operator σ : T → T and graininess function µ : T → [0,∞) are defined by σ(t) =
inf {s ∈ T : s > t} and µ(t) = σ(t) − t for t ∈ T, respectively. In this definition, we put inf φ = sup T,
where φ is an empty set. A half closed interval on T is given by

[a,b)T = {t ∈ T: a ≤ t < b} or (a,b]T = {t ∈ T: a < t ≤ b} .
Open and closed intervals can be defined similarly in [4], [19].

Let A be the family of all left closed and right open intervals on T of the form [a,b)T and m̃ : A →
[0,∞) be a set function on A such that m̃ ([a,b)T) = b−a. Then, it is known that m̃ is a countably
additive measure on A. Now, the Caratheodory extension of the set function m̃ associated with family
A is said to be the Lebesque ∆-measure on T and is denoted by µ∆. In this case, it is known that
if a ∈ T−{max T} , then the single point set {a} is ∆-measurable and µ∆(a) = σ(a) −a. If a,b ∈ T
and a ≤ b, then µ∆ ((a,b)T) = b−σ(a) and µ∆ ([a,b)T) = b−a. If a,b ∈ T−{max T} and a ≤ b, then
µ∆ ((a,b]T) = σ(b) −σ(a) and µ∆ ([a,b])T) = σ(b) −a (see [38]).

Statistical convergence is applied to time scales for different purposes by various authors in the
literature. For instance, Seyyidoglu and Tan [35] gave some important notions such as ∆-convergence,
∆-Cauchy by using ∆-density an investigate their relations on T and, in the recent past, they explained
a generalization of statistical cluster and limit points [36]. Turan and Duman [38] introduced density
and statistical convergence of ∆-measurable real-valued functions defined on T. Furthermore, Altin
et al. [1] expressed m- and (λ,m)-uniform density of a set and m- and (λ,m)-uniform statistical
convergence on T. However, Yilmaz et al. [41] defined λ-statistical convergence on T. Turan and
Duman [39] defined lacunary sequence and lacunary statistical convergence on T. Now, we give a
generalization of their study in a different form where θ = {kt−t0+1} is a lacunary sequence on T.
Definition 1.1. Let Ω be a ∆-measurable subset of T and θ be a lacunary sequence. Then, we define
the set Ω (t,θ) by

Ω (t,θ) = {s ∈ (kt−2t0+1,kt−t0+1]T : s ∈ Ω} ,
for t ∈ T. In this case, the θ-density of Ω on T is denoted by

δθT (Ω) = lim
t→∞

µ∆ (Ω (t,θ))

µ∆ ((kt−2t0,kt−t0 ]T)
,



UNIFORM LACUNARY STATISTICAL CONVERGENCE ON TIME SCALES 101

provided that the above limit exists.

Definition 1.2. Let f : T → R be a ∆-measurable function and θ be a lacunary sequence. Then, f is
lacunary statistically convergent to a real number L on T if

lim
t→∞

µ∆ (s ∈ (kt−2t0+1,kt−t0+1]T : |f (s) −L| ≥ ε)
µ∆ ((kt−2t0,kt−t0 ]T)

= 0,

for ∀ε > 0 and t ∈ T. In this case, sθT- lim
t→∞

(f (t)) = L. The set of all lacunary statistical convergence

functions on T will be denoted by sθT. (kt−2t0+1,kt−t0+1] turns to (kr−1,kr] for t = r, t0 = 1 and T = N.
In this case, lacunary statistical convergence on time scales is reduced to classical lacunary statistical
convergence which is defined by Fridy and Orhan [16].

In this study, we will give notions of (θ,m)-uniform lacunary density of an arbitrary set, (θ,m)-
uniform lacunary statistical convergence and some properties of (θ,m)-lacunary statistically convergent
sequences on an arbitrary time scale. Before this, we recall some concepts about uniform density and
uniform statistical convergence in classical case to use in our main results. Uniformly density of an
arbitrary set was introduced by Raimi [32] as follows:

Definition 1.3. A subset E ⊂ N is uniformly dense if

u (E) = lim
n→∞

1

n

n∑
j=1

χE (j + m) = a,

or equivalently

lim
n→∞

1

n
|E ∩{m + 1, ...,m + n}| = a,

uniformly in m, where m = 0, 1, 2, ... and χE is characteristic function.

Subsequently, uniformly density was studied by Baláž and Šalát [2]. Later, m-uniform statistical
convergence is introduced by Nuray [30] in the following manner.

Definition 1.4. Let x = (xk) be a real or complex valued sequence. If

lim
n→∞

1

n
|{m ≤ k < n + m : |xk −L| ≥ ε}| = 0,

uniformly in m, x = (xk) is said to be m-uniform statistically convergent to L for all ε > 0.

Based on Definition 1.4, we can generalize m-uniform statistical convergence to lacunary type se-
quences as follows:

Definition 1.5. Let K ⊂ N and θ be a lacunary sequence. Then, we define the (θ,m)-uniform density
of K by

δmθ (K) = lim
r→∞

1

hr,m
|{kr−1+m < k ≤ kr+m : k ∈ K}| ,

uniformly in m ≥ 0, where hr,m = kr+m −kr+m−1.

Definition 1.6. A sequence x = (xk) is said to be (θ,m)-uniform lacunary statistically convergent to
a real number L if

lim
r→∞

1

hr,m
|{kr−1+m < k ≤ kr+m : |xk −L| ≥ ε}| = 0,

for all ε > 0, uniformly in m.

Above definitions are special cases of σ-statistical convergence and lacunary σ-statistical convergence
[34]. In the next section, we shall define above notions on time scale T.



102 YILMAZ, MOHIUDDINE, ALTIN AND KOYUNBAKAN

2. Main results

In this section, we define and study the (θ,m)-density, (θ,m)-uniform lacunary statistical conver-
gence and (θ,m)-uniform strongly p-lacunary summability on T, where θ = {kt−t0+m+1} is a lacunary
sequence for t ∈ T.

Definition 2.1. Let Ω be a ∆-measurable subset of T and θ be a lacunary sequence. Then, we can
define the set Ω (t,θ,m) by

Ω (t,θ,m) = {s ∈ (kt−2t0+m+1,kt−t0+m+1]T : s ∈ Ω} ,
for t ∈ T. In this case, (θ,m)-density of Ω on T is defined by

δ
θ,m
T (Ω) = limt→∞

µ∆ (Ω (t,θ,m))

µ∆ ((kt−2t0+m,kt−t0+m]T)
, (2.1)

provided that the above limit exists.

Definition 2.2. Let f : T → R be a ∆-measurable function and θ be a lacunary sequence. Then, f is
(θ,m)-uniform lacunary statistically convergent to a real number L on T if

lim
t→∞

µ∆ (s ∈ (kt−2t0+m+1,kt−t0+m+1]T : |f (s) −L| ≥ ε)
µ∆ ((kt−2t0+m,kt−t0+m]T)

= 0, (2.2)

uniformly in m, for all ε > 0 and t ∈ T. In this case, sθ,mT - limt→∞ (f (t)) = L. The set of all (θ,m)-

uniform lacunary statistically convergent functions on T will be denoted by sθ,mT .

We remark that (kt−2t0+m+1,kt−t0+m+1] turns to (kr+m−1,kr+m] when t = r, t0 = 1 and T = N.
In this instance, (θ,m)-uniform lacunary statistical convergence on time scales is reduced to classical
(θ,m)-uniform lacunary statistical convergence which is given by Definition 1.6. This shows that our
results are generalizations of classical results.

Proposition 2.1. Let θ be a lacunary sequence. If f,g : T → R with sθ,mT - limt→∞f (t) = L1 and

s
θ,m
T - limt→∞

g (t) = L2, then the following statements hold:

(i) s
θ,m
T - limt→∞

(f (t) + g (t)) = L1 + L2,

(ii) s
θ,m
T - limt→∞

(cf (t)) = cL1 (c ∈ R) .

However, m-uniform statistical convergence on T was first defined by Altin et al. [1] in the following
way.

Definition 2.3. Let f : T → R be a ∆-measurable function. Then, f is m-uniform statistically
convergent to a real number L on T if

lim
t→∞

µ∆ (s ∈ [m + t0 − 1, t + m) : |f (s) −L| ≥ ε)
µ∆ ([m + t0 − 1, t + m)T)

= 0,

for all ε > 0 and uniformly in m. In this case, smT - lim
t→∞

(f (t)) = L. The set of all m-uniform statistically

convergent functions on T is denoted by smT .

Note that above Definition 2.3 is a generalization of Definition 1.4. Now we can give some inclusion

relations between smT , s
θ,m
T and s

θ
T.

Theorem 2.1. Let θ = {kt−t0+m+1} be a lacunary sequence for t ∈ T uniformly in m. Then,

(i) s
θ,m
T ⊂ s

m
T if lim supt

(
kt−t0+m+1
kt−2t0+m+1

)
< ∞,

(ii) smT ⊂ s
θ
T if lim inft

(
kt−t0+m+1
kt−2t0+m+1

)
> 1,

(iii) smT = s
θ
T if 1 < lim inft

(
kt−t0+m+1
kt−2t0+m+1

)
< lim supt

(
kt−t0+m+1
kt−2t0+m+1

)
< ∞.

Proof. It can be proved by using a similar approach to Theorem 3.3 of [31]. �



UNIFORM LACUNARY STATISTICAL CONVERGENCE ON TIME SCALES 103

The definition of strongly p-Cesàro summability on T was given by Turan and Duman [38] in the
following manner.

Definition 2.4. Let f : T → R be a ∆-measurable function and 0 < p < ∞. Then, f is strongly
p-Cesàro summable on T if there exists some L ∈ R such that

lim
t→∞

1

µ∆ ([t0, t]T)

∫
[t0,t]T

|f (s) −L|p ∆s = 0.

The set of all strongly p-Cesàro summable functions on T is denoted by [Wp]T .

The measure theory on time scales was first constructed by Guseinov [19] and Lebesque ∆-integral
on time scales introduced by Cabada and Vivero [6]. Now, we introduce m-uniform strongly p-
summablility and (θ,m)-uniform strongly p-lacunary summability of a ∆-measurable function and
establish some results.

Definition 2.5. Let f : T → R be a ∆-measurable function and 0 < p < ∞. Then, f is m-uniform
strongly p-summable on T if there exists some L ∈ R such that

lim
t→∞

1

µ∆ ([m + t0 − 1, t + m)T)

∫
[m+t0−1,t+m)T

|f (s) −L|p ∆s = 0.

In this case,
[
Wmp

]
T - lim f (s) = L. The set of all m-uniform strongly p-summable functions on T will

be denoted by
[
Wmp

]
T .

Definition 2.6. Let f : T → R be a ∆-measurable function and let θ be a lacunary sequence. Assume
also that 0 < p < ∞. Then, f is (θ,m)-uniform strongly p-lacunary summable on T if there exists
some L ∈ R such that

lim
t→∞

1

µ∆ ((kt−2t0+m,kt−t0+m]T)

∫
(kt−2t0+m+1,kt−t0+m+1]T

|f (s) −L|p ∆s = 0.

In that case,
[
Wmθp

]
T

- lim f (s) = L. The set of all (θ,m)-uniform strongly p-lacunary summable

functions on T will be denoted by
[
Wmθp

]
T
.

Lemma 2.1. Let f : T → R be a ∆-measurable function, θ be a lacunary sequence and

Ω (t,θ,m) = {s ∈ (kt−2t0+m+1,kt−t0+m+1]T : |f (s) −L| ≥ ε} ,

for all ε > 0. Thus, we have

µ∆ (Ω (t,θ,m)) ≤
1

ε

∫
Ω(t,θ,m)

|f (s) −L|∆s ≤
1

ε

∫
(kt−2t0+m+1,kt−t0+m+1]T

|f (s) −L|∆s,

uniformly in m.

Proof. It can be proved by using similar way with in [38]. �

Theorem 2.2. Let f : T → R be a ∆-measurable function and let θ be a lacunary sequence. Asume
also that 0 < p < ∞ and L ∈ R. Then,

(i) If f is (θ,m)-uniform strongly p-lacunary summable to L, then s
θ,m
T - limt→∞

(f (t)) = L.

(ii) If s
θ,m
T - limt→∞

(f (t)) = L and f is a bounded function, then f is (θ,m)-uniform strongly p-

lacunary summable to L.

Proof. (i) Suppose f is (θ,m)-uniform strongly p-lacunary summable to L. For given ε > 0, let

Ω (t,θ,m) = {s ∈ (kt−2t0+m+1,kt−t0+m+1]T : |f (s) −L| ≥ ε}



104 YILMAZ, MOHIUDDINE, ALTIN AND KOYUNBAKAN

on T. Then, it follows

εpµ∆ (Ω (t,θ,m)) ≤
∫

(kt−2t0+m+1,kt−t0+m+1]T

|f (s) −L|p ∆s.

from lemma 2.1. Dividing this inequality by µ∆ ((kt−2t0+m,kt−t0+m]T) and taking limit as t →∞, we
obtain

lim
t→∞

µ∆ (Ω (t,θ,m))

µ∆ ((kt−2t0+m,kt−t0+m]T)

≤
1

εp
lim
t→∞

1

µ∆ ((kt−2t0+m,kt−t0+m]T)

∫
(kt−2t0+m+1,kt−t0+m+1]T

|f (s) −L|p ∆s = 0,

which yields that s
θ,m
T - limt→∞

(f (t)) = L.

(ii) Suppose f is bounded and (θ,m)-uniform lacunary statistically convergent to L on T. Then,
there exists a positive number M such that |f (s)| ≤ M for all s ∈ T, and also

lim
t→∞

µ∆ (Ω (t,θ,m))

µ∆ ((kt−2t0+m,kt−t0+m]T)
= 0, (2.3)

where Ω (t,θ,m) as defined before. Since∫
(kt−2t0+m+1,kt−t0+m+1]T

|f (s) −L|p ∆s =
∫

Ω(t,θ,m)

|f (s) −L|p ∆s

+

∫
(kt−2t0+m+1,kt−t0+m+1]T/Ω(t,θ,m)

|f (s) −L|p ∆s

≤ (M + |L|)p
∫

Ω(t,θ,m)

∆s + εp
∫

(kt−2t0+m+1,kt−t0+m+1]T

∆s

= (M + |L|)p µ∆ (Ω (t,θ,m))
+εpµ∆ ((kt−2t0+m+1,kt−t0+m+1]T) ,

we obtain

lim
t→∞

1

µ∆ ((kt−2t0+m,kt−t0+m]T)

∫
(kt−2t0+m+1,kt−t0+m+1]T

|f (s) −L|p ∆s

≤ (M + |L|)p lim
t→∞

µ∆ (Ω (t,θ,m))

µ∆ ((kt−2t0+m,kt−t0+m]T)
+ εp. (2.4)

Since ε is an arbitrary, the proof follows from (2.3) and (2.4). �

Theorem 2.3. Let θ = {kt−t0+m+1} be a lacunary sequence for t ∈ T. Then

(i)
[
Wmθp

]
T
⊂
[
Wmp

]
T if lim supt

(
kt−t0+m+1
kt−2t0+m+1

)
< ∞,

(ii)
[
Wmp

]
T ⊂

[
Wmθp

]
T

if lim inft

(
kt−t0+m+1
kt−2t0+m+1

)
> 1,

(iii)
[
Wmp

]
T =

[
Wmθp

]
T

if 1 < lim inft

(
kt−t0+m+1
kt−2t0+m+1

)
< lim supt

(
kt−t0+m+1
kt−2t0+m+1

)
< ∞.

Proof. We can prove by using similar techniques to Theorem 2.2, Theorem 2.3 and Theorem 2.4 of [31]
in case of p = 1. �



UNIFORM LACUNARY STATISTICAL CONVERGENCE ON TIME SCALES 105

3. Conclusion

In this study, we defined the concept of (θ,m)-uniform lacunary density, (θ,m)-uniform lacunary
statistical convergence and (θ,m)-uniform strongly p-lacunary summability on T. We emphasize that
the results that we obtained are more general than classical results mentioned in the theory of m-
uniform statistical convergence. For example, Definition 1.6 is a generalization of the Definition 1.4
which is given by Nuray [30] to the lacunary type sequences. We firstly defined (θ,m)-uniform lacu-
nary statistical convergence in classical case to define it on T. Then, we generalized this definition
into T. Furthermore, we defined m-uniform strongly p-summable functions and m-uniform statistical
convergence on T by considering curicial results of Turan and Duman [38].

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1Firat University, Department of Mathematics, 23119, Elazıg, Turkey

2Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science,

King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

∗Corresponding author: mohiuddine@gmail.com


	1. Introduction and preliminaries
	2. Main results
	3. Conclusion
	References