A Tauberian Theorem for Double Cesàro Summability Method

Abstract

We have generalized Littlewood Tauberian theorems for summability of double sequences by using oscillating behavior and de la Vallée-Poussin mean. Further, the generalization of summability from summability is given as corollaries which were earlier established by the authors.

1. Introduction

Let be a double real sequence. We define , , and .

A double sequence is said to be bounded if there exists a finite real number such that . Let us write (see [1]),Then, we say that a double sequence is summable to if converges to , as . Similarly, we say that it is summable to if converges to as and summable to if converges to as . For all nonnegative integers and , we define as follows:

A double sequence is said to be summable to if converges to . If and , then summability reduces to summability. Again, if and , then summability reduces to summability. Further, if and , then summability reduces to summability. Here, Cesàro summability of refers to summability of . It may also be noted that the convergence of a double sequence implies the Cesàro summability of , but the converse is not generally true.

For example, consider a function : the sequence , which is the sequence of coefficients in the Taylor series expansion of the function about origin, is Cesàro summable but not convergent.

For the proof of the converse part, certain conditions are presented in terms of oscillatory behavior of double sequence .

Let us define aswhere(see [2]).

Moreover, in analogy to Kronecker identity for a single sequence, we can writeas the mean of the sequence and the mean of the sequence “,” respectively.

Further, as the sequence is the mean of the sequence , the sequence is summable to whenever converges to as . For all nonnegative integers and , let us define as follows:

The sequence is said to be summable to if converges to as . In particular, if and , then summability reduces to summability. Again, if and , then summability reduces to summability. Further, if and , then summability reduces to summability.

Then, the de la Vallée-Poussin mean of double real sequence is defined byfor sufficiently large nonnegative integers and , andfor sufficiently large nonnegative integers and .

A single sequence is slowly oscillating [3] if A double sequence is slowly oscillating [4] if

In an earlier paper by Jena et. al. [5], a proof of the generalized Littlewood Tauberian theorem by Cesàro summability method has been established. For a proof of Littlewood Tauberian theorem differently, the paper of Çanak and Totur [6] and Çanak [7–9] can be referred to. Also, a similar result was introduced earlier by Çanak [10] under the consideration of improper integral. Recently, Totur [4] has introduced Littlewood Tauberian theorem by mean for double real sequence.

In the proposed paper, with certain novelty, we have generalized it for summability of a double real sequence defined in (4).

2. Known Results

Theorem 1 (see [4]). If the sequence is summable to and is slowly oscillating (in the sense of ), then .

Corollary 2 (see [5]). If the sequence is summable to and is slowly oscillating, then .

Theorem 3 (see [4]). If the sequence is summable to and is slowly oscillating, then .

Corollary 4 (see [5]). If the sequence is summable to and is slowly oscillating, then .

3. Main Result

Theorem 5. If is summable to and is slowly oscillating, then .

To prove the above theorem, we need the help of the following lemmas.

Lemma 6. A double sequence is slowly oscillating if and only if is slowly oscillating and bounded.

Proof. Let be slowly oscillating. Initially, let us show that .
We have by definition of slow oscillation, for , and let us rewrite the finite sum as the series .
Clearly, where .
Consequently, we have Since is slowly oscillating, is slowly oscillating.
To prove the converse part, consider to be bounded and slowly oscillating. Now, the boundedness of implies that is slowly oscillating. Further, is slowly oscillating, so, by Kronecker identity (5), is slowly oscillating.
This completes the proof of Lemma 6.

Next, we represent the difference under two different cases in the following lemma.

Lemma 7 (see [1]). Let be a sequence of real numbers with sufficiently large; then one has the following: (i)For ,(ii)For ,

Proof of Theorem 5. Let be slowly oscillating; then, is slowly oscillating (by Lemma 6). Further, since is summable to , by Theorem 1,Next, from the definition,Clearly, (17) and (18) imply that is summable to . Again, is also slowly oscillating (by Lemma 6).
Thus, by Theorem 1, we haveContinuing in this way, we get .

Remark 8. If and , then summability reduces to summability. Again, for and , summability reduces to summability and, consequently, the following corollary is generated from the main result.

Corollary 9 (see [5]). If is or summable to and is slowly oscillating, then .

Theorem 10. If is summable to and is slowly oscillating, then .

Proof. As is slowly oscillating, setting in place of , is slowly oscillating by Lemma 6. Again, as is summable to , by Theorem 3, we haveBy definition, From (20) and (21), we have that is summable to . Again, by Lemma 6, since is slowly oscillating, we have (by Theorem 3). Continuing in this way, we get .

Remark 11. If and , then summability reduces to summability. Again, for and , summability reduces to summability and consequently the following corollaries are generated from the main result.

Corollary 12 (see [5]). If is summable to and is slowly oscillating, then .

Corollary 13 (see [5]). If is summable to and is slowly oscillating, then .

Competing Interests

The authors declare that there are no competing interests regarding publication of this paper.