Classical Tauberian Theorems for the (C; 1; 1) Summability Method

Totur, Ümi̇t

doi:10.2478/aicu-2014-0010

Cited by 12 publications

(17 citation statements)

References 4 publications

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“…A double sequence u = (u mn ) is said to be bounded if there exists a real number C > 0 such that |u mn | ≤ C for all nonnegative integers m and n. Note that a P-convergent double sequence need not be bounded (see [27] for details).…”

Section: The Statistical (C 1 1) Summability and Classical Tauberiamentioning

confidence: 99%

“…The converse of this implication is not true in general. An example that the converse implication does not hold is given by Totur [27]. The first study regarding classical Tauberian theorems for the (C, 1, 1) summability method is obtained by Knopp [14].…”

Section: The Statistical (C 1 1) Summability and Classical Tauberiamentioning

confidence: 99%

“…Therefore, we do not give the details of the proof. In addition, Totur [27] obtained Tauberian theorems of classical type by imposing different type of conditions such as one-sided boundedness and slow oscillation on the sequences V (10)…”

Section: The Statistical (C 1 1) Summability and Classical Tauberiamentioning

confidence: 99%

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Tauberian theorems for the statistical convergence and the statistical (C,1,1) summability

Totur

Çanak

2018

Filomat

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Every P-convergent double sequence is statistically convergent and every bounded statistically convergent sequence is statistical (C, 1, 1) summable. The converse of these implications are not always true. Theorems on which conditioned converses are searched are known as Tauberian theorems. Inspired by the convergence to zero of the difference sequence between a sequence and its arithmetic means in the single sequence case, we obtain Tauberian theorems for the statistical convergence and statistical (C, 1, 1) summability method by imposing some conditions on the difference sequence between a double sequence and its different arithmetic means. n k=1 k∆u k , is called the Kronecker identity.

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Section: The Statistical (C 1 1) Summability and Classical Tauberiamentioning

confidence: 99%

Section: The Statistical (C 1 1) Summability and Classical Tauberiamentioning

confidence: 99%

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Tauberian theorems for the statistical convergence and the statistical (C,1,1) summability

Totur

Çanak

2018

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“…Tauberian theorems in which Cesàro summability follows from Abel summability have a long tradition, which goes back to Hardy and Littlewood [14,11]. Such results have also received much attention in recent times, e.g., [1,15]. Actually, Pati and Ç anak et al have made extensive use of Tauberian conditions involving the Cesàro means of nc n , such as (1.3), in the study of Tauberian theorems for the so called (A)(C, α) summability.…”

Section: Introductionmentioning

confidence: 99%

Distributional versions of littlewood’s Tauberian theorem

Estrada

Vindas

2013

Czech Math J

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Abstract. We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We apply these Tauberian results to deduce a number of Tauberian theorems for power series where Cesàro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series. IntroductionA century ago, Littlewood obtained his celebrated extension of Tauber's theorem [22,14]. Littlewood's Tauberian theorem states that if the series ∞ n=0 c n is Abel summable to the number a, namely, the power series ∞ n=0 c n r n has radius of convergence at least 1 andand if the Tauberian hypothesisis satisfied, then the series is actually convergent, ∞ n=0 c n = a. The result was later strengthened by Hardy and Littlewood in [9,10] to an one-sided version. They showed that the condition (1.2) can be relaxed to the weaker one nc n = O L (1), i.e., there exists C > 0 such that −C < nc n .

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“…Finally, we mention that Pati [15] and Ç anak et al [3] have recently made use of Tauberian conditions involving Cesàro average versions of (1.6) in the study of Tauberian theorems for the so-called (A)(C, α) summability.…”

Section: Introductionmentioning

confidence: 99%