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Journal of Mathematics
Volume 2013 (2013), Article ID 573756, 4 pages
Lacunary Statistical Convergence of Sequences of Real-Valued Functions
Department of Secondary Science and Mathematics Education, Firat University, 23119 Elaziğ, Turkey
Received 10 September 2012; Accepted 26 November 2012
Academic Editor: Peijun Guo
Copyright © 2013 A. Gökhan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
The idea of the statistical convergence of sequence of real and complex numbers was introduced by Fast  and also independently by Buck  and Shoenberg . Gökhan and Güngör  defined pointwise statistical convergence of sequences of real-valued functions by saying that or on if and only if for every ,
By a lacunary sequence we mean an increasing integer sequence such that and as . Throughout this paper the intervals determined by will be denoted by and the ratio will be abbreviated by .
Quite recently Gökhan et al.  defined the strongly lacunary convergence of sequences of real-valued functions in the following way.
For any lacunary sequence , a sequence of functions is said to be strongly lacunary convergent to on the set if there exists such that for every . We denote this symbolically by writing on .
2. Lacunary Statistical Convergence
Now, we introduce the concepts of the lacunary statistical convergence of sequences of real valued functions.
Definition 1. Let be a lacunary sequence, and the sequence is -lacunary statistically convergent to on provided that for every ,
In this case we write or on .
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 on on and on under certain restrictions on .
Theorem 2. Let be a lacunary sequence; then(i) implies on ;(ii) for every and implies on ;(iii)let for every . Then if and only if on .
Proof. (i) If and on we can write
which yields the result.
(ii) Suppose that and on , say for all and every . Given we get for every , from which the result follows.
Let be given and define to be , at the first integers in , and otherwise on . Note that is not bounded. We have, for every , that is, . On the other hand, on ; hence .
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 , implies on if and only if .
Proof. Suppose first that ; then there exists a such that for sufficiently large , which implies that
If on , then for every and for sufficiently large , we have
This proves the sufficiency.
Conversely, suppose that . Proceeding as in Lemma 2.1  we can select a subsequence of the lacunary sequence such that where .
Now define a bounded sequence by if for some and otherwise on . It is shown in Lemma 2.1  that is not strongly lacunarily convergent on , but is strongly Cesáro summability. Theorem 2(ii) implies that is not lacunarily statistically convergent, but it follows from Theorem 2.3. (i) of  that is pointwise statistically convergent on . Hence we obtain that is pointwise statistical convergent on , but it is not lacunarily statistically convergent on . This contradicts the assumption, whence .
Lemma 4. For any lacunary sequence , on implies on if and only if .
Proof. If , then there is an such that for all . Suppose that on , and let
By (4) given , there is an such that
Now let for every and let be any integer satisfying ; then we can write for every , and the sufficiency follows immediately.
Conversely, suppose that . Following the idea in Lemma 2.2  we can select a subsequence of the lacunary sequence such that , and define a bounded sequence by if for some , and otherwise on . It is shown in Lemma 2.2  that on but on . By Theorem 2(i) we conclude that on but Theorem 2.3 of  implies that on . Hence does not imply on .
Combining Lemmas 3 and 4 we get the following.
Theorem 5. Let be a lacunary sequence and let be a sequence of functions on a set ; then and if and only if
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 be a lacunary sequence and let be a sequence of functions on a set . The sequence is said to be a lacunary statistical Cauchy sequence if there is a subsequence of such that for each , on , and for every
Theorem 7. The sequence is lacunarily statistically convergent if and only if is a lacunary statistical Cauchy sequence.
Proof. Let on , and write
for each . Hence, for each , for every and
Choose such that implies for every . Next choose so that implies for every . Then for each satisfying , choose such that for every . In general, choose such that implies for every . Then for all satisfying , choose for every , that is: for every . Hence, we get for every , and (20) implies that for every . Furthermore, we have, for every , Using the assumption that on and on , we infer (17), whence is a lacunary statistical Cauchy sequence on .
Conversely, suppose that is a lacunary statistical Cauchy sequence. For every , we have from which it follows that on .
It is know that several different types of convergence may define sequences of which trems are real-valued functions having a common domain on the real line . The present paper gives the lacunary statistical convergence of a sequence of real-valued functions 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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