Lacunary Statistical Convergence of Sequences of Real-Valued Functions

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.

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−lim𝑓_𝑘(𝑥)=𝑓(𝑥) or 𝑓_𝑘−−→^st.𝑓 on 𝑆 if and only if for every 𝜀>0, lim_𝑛→∞1𝑛𝑘≤𝑛∶𝑓_𝑘(𝑥)−𝑓(𝑥)≥𝜀forevery𝑥∈𝑆=0. (1)

By a lacunary sequence we mean an increasing integer sequence 𝜃=(𝑘_𝑟) such that 𝑘₀=0 and ℎ_𝑟=𝑘_𝑟−𝑘_𝑟−1→∞ as 𝑟→∞. Throughout this paper the intervals determined by 𝜃 will be denoted by 𝐼_𝑟=(𝑘_𝑟−1,𝑘_𝑟] and the ratio 𝑘_𝑟/𝑘_𝑟−1 will be abbreviated by 𝑞_𝑟.

Quite recently Gökhan et al. [5] defined the strongly lacunary convergence of sequences of real-valued functions in the following way.

For any lacunary sequence 𝜃=(𝑘_𝑟), a sequence of functions (𝑓_𝑘) is said to be strongly lacunary convergent to 𝑓 on the set 𝑆⊂ℝ if there exists 𝑓(𝑥) such that 1ℎ_𝑟𝐼𝑟𝑓_𝑘(𝑥)−𝑓(𝑥)→0 (2) for every 𝑥∈𝑆. We denote this symbolically by writing 𝑓_𝑘−−→^𝑁𝜃𝑓 on 𝑆.

Fridy and Orhan [6] defined lacunary statistical convergence by saying that 𝑆_𝜃−lim𝑥=𝐿 or 𝑥_𝑘→𝐿(𝑆_𝜃) if and only if for every 𝜀>0, lim_𝑟1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑥_𝑘−𝐿≥𝜀=0. (3) 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. Let 𝜃 be a lacunary sequence, and the sequence (𝑓_𝑘) is 𝜃-lacunary statistically convergent to 𝑓(𝑥) on 𝑆 provided that for every 𝜀>0, lim_𝑟1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑓_𝑘(𝑥)−𝑓(𝑥)≥𝜀forevery𝑥∈𝑆=0. (4)
In this case we write 𝑆_𝜃−lim𝑓_𝑘(𝑥)=𝑓(𝑥) or 𝑓_𝑘−−→^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 study 𝑓_𝑘−−→^st.𝑓 on 𝑆⇒𝑓_𝑘−−→^𝑆𝜃𝑓 on 𝑆 and 𝑓_𝑘−−→^𝑆𝜃𝑓⇒𝑓_𝑘−−→^st.𝑓 on 𝑆 under certain restrictions on 𝜃=(𝑘_𝑟).

Theorem 2. Let 𝜃=(𝑘_𝑟) be a lacunary sequence; then(i)𝑓_𝑘−−→^𝑁𝜃𝑓 implies 𝑓_𝑘−−→^𝑆𝜃𝑓 on 𝑆;(ii)sup_𝑘|𝑓_𝑘(𝑥)|<∞ for every 𝑥∈𝑆 and 𝑓_𝑘−−→^𝑆𝜃𝑓 implies 𝑓_𝑘−−→^𝑁𝜃𝑓 on 𝑆;(iii)let sup_𝑘|𝑓_𝑘(𝑥)|<∞ for every 𝑥∈𝑆. Then 𝑓_𝑘−−→^𝑆𝜃𝑓 if and only if 𝑓_𝑘−−→^𝑁𝜃𝑓 on 𝑆.

Proof. (i) If 𝜀>0 and 𝑓_𝑘−−→^𝑁𝜃𝑓 on 𝑆 we can write _{𝑘∈𝐼𝑟}𝑓_𝑘(𝑥)(𝑥≥−𝑓)_{𝑘∈𝐼𝑟𝑓𝑘(𝑥)(𝑥−𝑓)≥𝜀}𝑓_𝑘(𝑥)−𝑓(𝑥)≥𝜀𝑘∈𝐼_𝑟∶𝑓_𝑘(𝑥)−𝑓(𝑥)≥𝜀forevery𝑥∈𝑆 (5) which yields the result.
(ii) Suppose that 𝑓−−→_𝑘^𝑆𝜃𝑓 and sup_𝑘|𝑓_𝑘(𝑥)|<∞ on 𝑆, say 𝑓_𝑘(𝑥)−𝑓(𝑥)≤𝑀_𝑥 (6) for all 𝑘 and every 𝑥∈𝑆. Given 𝜀>0 we get 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓_𝑘(𝑥)(𝑥=1−𝑓)ℎ_{𝑟𝑘∈𝐼𝑟𝑓𝑘(𝑥)(𝑥−𝑓)≥𝜀}𝑓_𝑘(𝑥)(𝑥+1−𝑓)ℎ_{𝑟𝑘∈𝐼𝑟𝑓𝑘(𝑥)(𝑥−𝑓)<𝜀}𝑓_𝑘(𝑥)(𝑥≤𝑀−𝑓)_𝑥ℎ_𝑟𝑘∈𝐼_𝑟∶𝑓_𝑘(𝑥)−𝑓(𝑥)≥𝜀forevery𝑥∈𝑆+𝜀, (7) for every 𝑥∈𝑆, from which the result follows.
Let 𝜃 be given and define 𝑓_𝑘(𝑥) to be |𝑥|,  2|𝑥|,…,[ℎ_𝑟]⋅|𝑥| at the first [ℎ_𝑟] integers in 𝐼_𝑟, and 𝑥_𝑘=0 otherwise on ℝ. Note that (𝑓_𝑘) is not bounded. We have, for every 𝜀>0, 1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑓_𝑘(𝑥)=−0≥𝜀forevery𝑥∈ℝℎ_𝑟ℎ_𝑟→0,as𝑟→∞ (8) that is, 𝑓−−→_𝑘^𝑆𝜃𝑓=0. On the other hand, 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓_𝑘(𝑥)=1−0ℎ_𝑟ℎ_𝑟ℎ_𝑟+1⋅|𝑥|2→|𝑥|2≠0 (9) on ℝ−{0}; hence 𝑓↛_𝑘^𝑁𝜃𝑓=0.
We remark that the example given in (ii) shows that the boundedness condition cannot be omitted from the hypothesis of Theorem 2(ii).
(iii) This is an immediate consequence of (i) and (ii).

Lemma 3. For any lacunary sequence 𝜃, 𝑓−−→_𝑘^𝑠𝑡.𝑓 implies 𝑓−−→_𝑘^𝑆𝜃𝑓 on 𝑆 if and only if  liminf_𝑟𝑞_𝑟>1.

Proof. Suppose first that liminf_𝑟𝑞_𝑟>1; then there exists a 𝛿>0 such that 𝑞_𝑟≥1+𝛿 for sufficiently large 𝑟, which implies that ℎ_𝑟𝑘_𝑟≥𝛿.1+𝛿 (10) If 𝑓−−→_𝑘^st.𝑓 on 𝑆, then for every 𝜀>0 and for sufficiently large 𝑟, we have 1𝑘_𝑟𝑘≤𝑘_𝑟∶𝑓_𝑘(𝑥)(𝑥≥1−𝑓)≥𝜀forevery𝑥∈𝑆𝑘_𝑟𝑘∈𝐼_𝑟∶𝑓_𝑘(𝑥)(𝑥≥𝛿−𝑓)≥𝜀forevery𝑥∈𝑆11+𝛿ℎ_𝑟𝑘∈𝐼_𝑟∶𝑓_𝑘(𝑥)−𝑓(𝑥)≥𝜀forevery𝑥∈𝑆. (11) This proves the sufficiency.
Conversely, suppose that liminf_𝑟𝑞_𝑟>1. Proceeding as in Lemma  2.1 [5] we can select a subsequence (𝑘_𝑟(𝑗)) of the lacunary sequence 𝜃=(𝑘_𝑟) such that 𝑘_𝑟(𝑗)𝑘_{𝑟(𝑗)−1}1<1+𝑗,𝑘_{𝑟(𝑗)−1}𝑘_{𝑟(𝑗−1)}>𝑗, (12) where 𝑟(𝑗)≥𝑟(𝑗−1)+2.
Now define a bounded sequence (𝑓_𝑘) by 𝑓_𝑖(𝑥)=𝑥 if 𝑖∈𝐼_𝑟(𝑗) for some 𝑗=1,2,… and 𝑓_𝑖(𝑥)=0 otherwise on ℝ. It is shown in Lemma  2.1 [5] that (𝑓_𝑘) is not strongly lacunarily convergent on ℝ, but (𝑓_𝑘) is strongly Cesáro summability. Theorem 2(ii) implies that (𝑓_𝑘) is not lacunarily statistically convergent, but it follows from Theorem  2.3. (i) of [5] that (𝑓_𝑘) is pointwise statistically convergent on 𝑆. Hence we obtain that (𝑓_𝑘) is pointwise statistical convergent on 𝑆, but it is not lacunarily statistically convergent on 𝑆. This contradicts the assumption, whence liminf_𝑟𝑞_𝑟>1.

Lemma 4. For any lacunary sequence 𝜃, 𝑓−−→_𝑘^𝑆𝜃𝑓 on 𝑆 implies 𝑓−−→_𝑘^𝑠𝑡.𝑓 on 𝑆 if and only if  limsup_𝑟𝑞_𝑟<∞.

Proof. If limsup_𝑟𝑞_𝑟<∞, then there is an 𝐻>0 such that 𝑞_𝑟<𝐻 for all 𝑟. Suppose that 𝑓−−→_𝑘^𝑆𝜃𝑓 on 𝑆, and let 𝑁_𝑟,𝑥=𝑘∈𝐼_𝑟∶𝑓_𝑘(𝑥)−𝑓(𝑥)≥𝜀forevery𝑥∈𝑆. (13) By (4) given 𝜀>0, there is an 𝑟₀∈ℕ such that 𝑁_𝑟,𝑥ℎ_𝑟<𝜀forall𝑟>𝑟₀andevery𝑥∈𝑆. (14)
Now let 𝑀_𝑥={max𝑁_𝑟,𝑥∶1≤𝑟≤𝑟₀} for every 𝑥∈𝑆 and let 𝑛 be any integer satisfying 𝑘_𝑟−1<𝑛≤𝑘_𝑟; then we can write 1𝑛𝑘≤𝑛∶𝑓_𝑘(𝑥)(𝑥≤1−𝑓)≥𝜀forevery𝑥∈𝑆𝑘_𝑟−1𝑘≤𝑘_𝑟∶𝑓_𝑘(𝑥)(𝑥=1−𝑓)≥𝜀forevery𝑥∈𝑆𝑘_𝑟−1𝑁_1,𝑥+𝑁_2,𝑥+⋯+𝑁_𝑟0,𝑥+𝑁_𝑟0+1,𝑥+⋯+𝑁_𝑟,𝑥≤𝑀_𝑥𝑘_𝑟−1⋅𝑟₀+1𝑘_𝑟−1ℎ_𝑟0+1𝑁_𝑟0+1,𝑥ℎ_𝑟0+1+⋯+ℎ_𝑟𝑁_𝑟,𝑥ℎ_𝑟≤𝑟₀⋅𝑀_𝑥𝑘_𝑟−1+1𝑘_𝑟−1sup_𝑟>𝑟0𝑁_𝑟,𝑥ℎ_𝑟ℎ_𝑟0+1+⋯+ℎ_𝑟≤𝑟₀⋅𝑀_𝑥𝑘_𝑟−1𝑘+𝜀⋅^𝑟−𝑘_𝑟0𝑘_𝑟−1()≤𝑟by14₀⋅𝑀_𝑥𝑘_𝑟−1+𝜀⋅𝑞_𝑟≤𝑟₀⋅𝑀_𝑥𝑘_𝑟−1+𝜀𝐻 (15) for every 𝑥∈𝑆, and the sufficiency follows immediately.
Conversely, suppose that limsup_𝑟𝑞_𝑟=∞. Following the idea in Lemma  2.2 [5] we can select a subsequence (𝑘_𝑟(𝑗)) of the lacunary sequence 𝜃=(𝑘_𝑟) such that 𝑞_𝑟(𝑗)>𝑗, and define a bounded sequence by 𝑓_𝑖(𝑥)=𝑥 if 𝑘_{𝑟(𝑗)−1}<𝑖≤2𝑘_{𝑟(𝑗)−1} for some 𝑗=1,2,…, and 𝑓_𝑖(𝑥)=0 otherwise on ℝ. It is shown in Lemma  2.2 [5] that 𝑓−−→_𝑘^𝑁𝜃𝑓=0 on ℝ but 𝑓↛_𝑘^|𝜎1|𝑓 on ℝ. By Theorem 2(i) we conclude that 𝑓−−→_𝑘^𝑆𝜃𝑓=0 on ℝ but Theorem  2.3 of [5] implies that 𝑓↛_𝑘^st.𝑓=0 on ℝ. Hence 𝑓−−→_𝑘^𝑆𝜃𝑓=0 does not imply 𝑓−−→_𝑘^st.𝑓=0 on ℝ.
Combining Lemmas 3 and 4 we get the following.

Theorem 5. Let 𝜃 be a lacunary sequence and let (𝑓_𝑘) be a sequence of functions on a set 𝑆; then 𝑓−−→_𝑘^st.𝑓 and 𝑓−−→_𝑘^𝑆𝜃𝑓 if and only if 1<liminf_𝑟𝑞_𝑟≤limsup_𝑟𝑞_𝑟<∞. (16)

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. Let 𝜃=(𝑘_𝑟) be a lacunary sequence and let (𝑓_𝑘) be a sequence of functions on a set 𝑆. The sequence (𝑓_𝑘) is said to be a 𝜃 lacunary statistical Cauchy sequence if there is a subsequence (𝑓_{𝑘′(𝑟)}) of (𝑓_𝑘) such that 𝑘′(𝑟)∈𝐼_𝑟 for each 𝑟,  lim_𝑟𝑓_{𝑘′(𝑟)}(𝑥)=𝑓(𝑥) on 𝑆, and for every 𝜀>0lim_𝑟1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑓_𝑘(𝑥)−𝑓_{𝑘′(𝑟)}(𝑥)≥𝜀forevery𝑥∈𝑆=0. (17)

Theorem 7. The sequence (𝑓_𝑘) is lacunarily statistically convergent if and only if (𝑓_𝑘) is a lacunary statistical Cauchy sequence.

Proof. Let 𝑓−−→_𝑘^𝑆𝜃𝑓 on 𝑆, and write 𝐾_𝑥^(𝑗)=𝑘∈ℕ∶𝑓_𝑘(𝑥)(𝑥<1−𝑓)𝑗forevery𝑥∈𝑆 (18) for each 𝑗∈ℕ. Hence, for each 𝑗, 𝐾_𝑥^(𝑗)⊇𝐾_𝑥^(𝑗+1) for every 𝑥∈𝑆 and 𝐾_𝑥^(𝑗)∩𝐼_𝑟ℎ_𝑟→1as𝑟→∞. (19)
Choose 𝑚(1) such that 𝑟≥𝑚(1) implies |𝐾_𝑥⁽¹⁾∩𝐼_𝑟|/ℎ_𝑟>0,thatis,𝐾_𝑥⁽¹⁾∩𝐼_𝑟≠𝜙 for every 𝑥∈𝑆. Next choose 𝑚(2)>𝑚(1) so that 𝑟≥𝑚(2) implies 𝐾_𝑥⁽²⁾∩𝐼_𝑟≠𝜙 for every 𝑥∈𝑆. Then for each 𝑟 satisfying 𝑚(1)≤𝑟<𝑚(2), choose 𝑘^′(𝑟)∈𝐼_𝑟 such that 𝑘^′(𝑟)∈𝐼_𝑟∩𝐾_𝑥⁽¹⁾,thatis,|𝑓_{𝑘′(𝑟)}(𝑥)−𝑓(𝑥)|<1 for every 𝑥∈𝑆. In general, choose 𝑚(𝑝+1)>𝑚(𝑝) such that 𝑟>𝑚(𝑝+1) implies 𝐼_𝑟∩𝐾_𝑥^(𝑝+1)≠𝜙 for every 𝑥∈𝑆. Then for all 𝑟 satisfying 𝑚(𝑝)≤𝑟<𝑚(𝑝+1), choose 𝑘^′(𝑟)∈𝐼_𝑟∩𝐾_𝑥^(𝑝) for every 𝑥∈𝑆, that is: 𝑓_{𝑘′(𝑟)}(𝑥)(𝑥<1−𝑓)𝑝 (20) for every 𝑥∈𝑆. Hence, we get 𝑘^′(𝑟)∈𝐼_𝑟 for every 𝑟, and (20) implies that lim_𝑟𝑓_{𝑘′(𝑟)}(𝑥)=𝑓(𝑥) for every 𝑥∈𝑆. Furthermore, we have, for every 𝜀>0, 1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑓_𝑘(𝑥)−𝑓_{𝑘′(𝑟)}(𝑥≤1)≥𝜀forevery𝑥∈𝑆ℎ_𝑟𝑘∈𝐼_𝑟∶𝑓_𝑘(𝑥)(𝑥≥𝜀−𝑓)2+1forevery𝑥∈𝑆ℎ_𝑟𝑘∈𝐼_𝑟∶𝑓_{𝑘′(𝑟)}(𝑥)(𝑥≥𝜀−𝑓)2forevery𝑥∈𝑆. (21) Using the assumption that 𝑓−−→_𝑘^𝑆𝜃𝑓 on 𝑆 and lim_𝑟𝑓_{𝑘′(𝑟)}(𝑥)=𝑓(𝑥) on 𝑆, we infer (17), whence (𝑓_𝑘(𝑥)) is a 𝜃 lacunary statistical Cauchy sequence on 𝑆.
Conversely, suppose that (𝑓_𝑘(𝑥)) is a 𝜃 lacunary statistical Cauchy sequence. For every 𝜀>0, we have 𝑘∈𝐼_𝑟∶𝑓_𝑘(𝑥)(𝑥≤−𝑓)≥𝜀forevery𝑥∈𝑆𝑘∈𝐼_𝑟∶𝑓_𝑘(𝑥)−𝑓_{𝑘′(𝑟)}(𝑥≥𝜀)2+forevery𝑥∈𝑆𝑘∈𝐼_𝑟∶𝑓_{𝑘′(𝑟)}(𝑥)(𝑥≥𝜀−𝑓)2forevery𝑥∈𝑆, (22) from which it follows that 𝑓−−→_𝑘^𝑆𝜃𝑓 on 𝑆.

4. Conclusion

It is know that several different types of convergence may define sequences (𝑓_𝑘) 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 (𝑓_𝑘) 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.