# Lacunary Statistical Convergence of Sequences of Real-Valued Functions

Authors: A. Gökhan
Journal: Journal of Mathematics
 (2013)

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## Abstract

We introduce the concepts of the lacunary statistical
convergence of sequences of real-valued functions. We also give the relation
between this convergence and strongly lacunary and pointwise statistical convergence. Furthermore we introduce the concept of a lacunary statistical Cauchy sequence for functional sequences and prove that it is equivalent to lacunary statistical convergence of sequences of real-valued functions.


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## Body

## 1. Introduction

The idea of the statistical convergence of sequence of real and complex numbers was introduced by Fast [1] and also independently by Buck [2] and Shoenberg [3]. Gökhan and Güngör [4] defined pointwise statistical convergence of sequences of real-valued functions by saying that st-limfk(x)=f(x) or fk→st.f on S if and only if for every ε>0,
(1)limn→∞1n|{k≤n:|fk(x)-f(x)|≥εforeveryx∈S}|=0.By a lacunary sequence we mean an increasing integer sequenceθ=(kr) such that k0=0 and hr=kr-kr-1→∞ as r→∞. Throughout this paper the intervals determined by θ will be denoted by Ir=(kr-1,kr] and the ratio kr/kr-1 will be abbreviated by qr.Quite recently Gökhan et al. [5] defined the strongly lacunary convergence of sequences of real-valued functions in the following way.For any lacunary sequenceθ=(kr), a sequence of functions (fk) is said to be strongly lacunary convergent to f on the set S⊂ℝ if there exists f(x) such that
(2)1hr∑Ir‍|fk(x)-f(x)|→0
for every x∈S. We denote this symbolically by writing fk→Nθf on S.Fridy and Orhan [6] defined lacunary statistical convergence by saying that Sθ-limx=L or xk→L(Sθ) if and only if for every ε>0,
(3)limr1hr|{k∈Ir:|xk-L|≥ε}|=0.
Furthermore, Fridy and Orhan [7] introduced the lacunary statistical Cauchy sequence.


## 2. Lacunary Statistical Convergence

Now, we introduce the concepts of the lacunary statistical convergence of sequences of real valued functions.Definition 1.
Letθ be a lacunary sequence, and the sequence (fk) is θ-lacunary statistically convergent to f(x) on S provided that for every ε>0,
(4)limr1hr|{k∈Ir:|fk(x)-f(x)|≥εforeveryx∈S}|=0.
In this case we writeSθ-limfk(x)=f(x) or fk→Sθf on S.
Now, we first give some relations between strongly lacunary convergence and lacunary stratistical convergence and show that they are equivalent for bounded sequences. We also studyfk→st.f on S⇒fk→Sθf on S and fk→Sθf⇒fk→st.f on S under certain restrictions on θ=(kr).Theorem 2.
Letθ=(kr) be a lacunary sequence; then(i)
fk→Nθf implies fk→Sθf on S;(ii)
supk|fk(x)|<∞ for every x∈S and fk→Sθf implies fk→Nθf on S;(iii)
letsupk|fk(x)|<∞ for every x∈S. Then fk→Sθf if and only if fk→Nθf on S.Proof.
(i) Ifε>0 and fk→Nθf on S we can write
(5)∑k∈Ir|fk(x)-f(x)|≥∑k∈Ir|fk(x)-f(x)|≥ε|fk(x)-f(x)|≥ε|{k∈Ir:|fk(x)-f(x)|≥εforeveryx∈S}|
which yields the result.
(ii) Suppose thatfk→Sθf and supk|fk(x)|<∞ on S, say
(6)|fk(x)-f(x)|≤Mx
for all k and every x∈S. Given ε>0 we get
(7)1hr∑k∈Ir‍|fk(x)-f(x)|=1hr∑k∈Ir|fk(x)-f(x)|≥ε|fk(x)-f(x)|+1hr∑k∈Ir|fk(x)-f(x)|<ε|fk(x)-f(x)|≤Mxhr|{k∈Ir:|fk(x)-f(x)|≥εforeveryx∈S}|+ε,
for every x∈S, from which the result follows.
Letθ be given and define fk(x) to be |x|,  2|x|,…,[hr]·|x| at the first [hr] integers in Ir, and xk=0 otherwise on ℝ. Note that (fk) is not bounded. We have, for every ε>0,
(8)1hr|{k∈Ir:|fk(x)-0|≥εforeveryx∈ℝ}|=[hr]hr→0,asr→∞
that is, fk→Sθf=0. On the other hand,
(9)1hr∑k∈Ir‍|fk(x)-0|=1hr[hr]([hr]+1)·|x|2→|x|2≠0
on ℝ-{0}; hence fk↛Nθf=0.
We remark that the example given in (ii) shows that the boundedness condition cannot be omitted from the hypothesis of Theorem2(ii).
(iii) This is an immediate consequence of (i) and (ii).Lemma 3.
For any lacunary sequenceθ, fk→st.f implies fk→Sθf on S if and only if  liminfrqr>1.Proof.
Suppose first thatliminfrqr>1; then there exists a δ>0 such that qr≥1+δ for sufficiently large r, which implies that
(10)hrkr≥δ1+δ.
If fk→st.f on S, then for every ε>0 and for sufficiently large r, we have
(11)1kr|{k≤kr:|fk(x)-f(x)|≥εforeveryx∈S}|≥1kr|{k∈Ir:|fk(x)-f(x)|≥εforeveryx∈S}|≥δ1+δ1hr|{k∈Ir:|fk(x)-f(x)|≥εforeveryx∈S}|.
This proves the sufficiency.
Conversely, suppose thatliminfrqr>1. Proceeding as in Lemma  2.1 [5] we can select a subsequence (kr(j)) of the lacunary sequence θ=(kr) such that
(12)kr(j)kr(j)-1<1+1j,kr(j)-1kr(j-1)>j,
where r(j)≥r(j-1)+2.
Now define a bounded sequence(fk) by fi(x)=x if i∈Ir(j) for some j=1,2,… and fi(x)=0 otherwise on ℝ. It is shown in Lemma  2.1 [5] that (fk) is not strongly lacunarily convergent on ℝ, but (fk) is strongly Cesáro summability. Theorem 2(ii) implies that (fk) is not lacunarily statistically convergent, but it follows from Theorem  2.3. (i) of [5] that (fk) is pointwise statistically convergent on S. Hence we obtain that (fk) is pointwise statistical convergent on S, but it is not lacunarily statistically convergent on S. This contradicts the assumption, whence liminfrqr>1.Lemma 4.
For any lacunary sequenceθ, fk→Sθf on S implies fk→st.f on S if and only if  limsuprqr<∞.Proof.
Iflimsuprqr<∞, then there is an H>0 such that qr<H for all r. Suppose that fk→Sθf on S, and let
(13)Nr,x=|{k∈Ir:|fk(x)-f(x)|≥εforeveryx∈S}|.
By (4) given ε>0, there is an r0∈ℕ such that
(14)Nr,xhr<εforallr>r0andeveryx∈S.
Now letMx={maxNr,x:1≤r≤r0} for every x∈S and let n be any integer satisfying kr-1<n≤kr; then we can write
(15)1n|{k≤n:|fk(x)-f(x)|≥εforeveryx∈S}|≤1kr-1|{k≤kr:|fk(x)-f(x)|≥εforeveryx∈S}|=1kr-1{N1,x+N2,x+⋯+Nr0,x+Nr0+1,x+⋯+Nr,x}≤Mxkr-1·r0+1kr-1{hr0+1Nr0+1,xhr0+1+⋯+hrNr,xhr}≤r0·Mxkr-1+1kr-1(supr>r0Nr,xhr){hr0+1+⋯+hr}≤r0·Mxkr-1+ε·kr-kr0kr-1by(14)≤r0·Mxkr-1+ε·qr≤r0·Mxkr-1+εH
for every x∈S, and the sufficiency follows immediately.
Conversely, suppose thatlimsuprqr=∞. Following the idea in Lemma  2.2 [5] we can select a subsequence (kr(j)) of the lacunary sequence θ=(kr) such that qr(j)>j, and define a bounded sequence by fi(x)=x if kr(j)-1<i≤2kr(j)-1 for some j=1,2,…, and fi(x)=0 otherwise on ℝ. It is shown in Lemma  2.2 [5] that fk→Nθf=0 on ℝ but fk↛|σ1|f on ℝ. By Theorem 2(i) we conclude that fk→Sθf=0 on ℝ but Theorem  2.3 of [5] implies that fk↛st.f=0 on ℝ. Hence fk→Sθf=0 does not imply fk→st.f=0 on ℝ.
Combining Lemmas3 and 4 we get the following.Theorem 5.
Letθ be a lacunary sequence and let (fk) be a sequence of functions on a set S; then fk→st.f and fk→Sθf if and only if
(16)1<liminfrqr≤limsuprqr<∞.


## 3. Cauchy Criterion

Now we introduce the lacunary statistical analog of the Cauchy convergence criterion which is, as we shall see, equivalent to the lacunary statistical convergence.Definition 6.
Letθ=(kr) be a lacunary sequence and let (fk) be a sequence of functions on a set S. The sequence (fk) is said to be a θ lacunary statistical Cauchy sequence if there is a subsequence (fk′(r)) of (fk) such that k′(r)∈Ir for each r,  limrfk′(r)(x)=f(x) on S, and for every ε>0(17)limr1hr|{k∈Ir:|fk(x)-fk′(r)(x)|≥εforeveryx∈S}|=0.Theorem 7.
The sequence(fk) is lacunarily statistically convergent if and only if (fk) is a lacunary statistical Cauchy sequence.Proof.
Letfk→Sθf on S, and write
(18)Kx(j)={k∈ℕ:|fk(x)-f(x)|<1jforeveryx∈S}
for each j∈ℕ. Hence, for each j, Kx(j)⊇Kx(j+1) for every x∈S and
(19)|Kx(j)∩Ir|hr→1asr→∞.
Choosem(1) such that r≥m(1) implies |Kx(1)∩Ir|/hr>0,thatis,Kx(1)∩Ir≠ϕ for every x∈S. Next choose m(2)>m(1) so that r≥m(2) implies Kx(2)∩Ir≠ϕ for every x∈S. Then for each r satisfying m(1)≤r<m(2), choose k′(r)∈Ir such that k′(r)∈Ir∩Kx(1),thatis,|fk′(r)(x)-f(x)|<1 for every x∈S. In general, choose m(p+1)>m(p) such that r>m(p+1) implies Ir∩Kx(p+1)≠ϕ for every x∈S. Then for all r satisfying m(p)≤r<m(p+1), choose k′(r)∈Ir∩Kx(p) for every x∈S, that is:
(20)|fk′(r)(x)-f(x)|<1p
for every x∈S. Hence, we get k′(r)∈Ir for every r, and (20) implies that limrfk′(r)(x)=f(x) for every x∈S. Furthermore, we have, for every ε>0,
(21)1hr|{k∈Ir:|fk(x)-fk′(r)(x)|≥εforeveryx∈S}|≤1hr|{k∈Ir:|fk(x)-f(x)|≥ε2foreveryx∈S}|+1hr|{k∈Ir:|fk′(r)(x)-f(x)|≥ε2foreveryx∈S}|.
Using the assumption that fk→Sθf on S and limrfk′(r)(x)=f(x) on S, we infer (17), whence (fk(x)) is a θ lacunary statistical Cauchy sequence on S.
Conversely, suppose that(fk(x)) is a θ lacunary statistical Cauchy sequence. For every ε>0, we have
(22)|{k∈Ir:|fk(x)-f(x)|≥εforeveryx∈S}|≤|{k∈Ir:|fk(x)-fk′(r)(x)|≥ε2foreveryx∈S}|+|{k∈Ir:|fk′(r)(x)-f(x)|≥ε2foreveryx∈S}|,
from which it follows that fk→Sθf on S.


## 4. Conclusion

It is know that several different types of convergence may define sequences(fk) of which trems are real-valued functions having a common domain on the real line ℝ. The present paper gives the lacunary statistical convergence of a sequence of real-valued functions (fk) and gives some relations between this convergence and strongly lacunary convergence and shows that they are equivalent for a bounded sequence of real-valued functions. Furthermore, the paper proves that this convergence is equivalent to lacunary statistical Cauchy sequence.



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