Some extended Tauberian theorems for [(A).sup.(k)] (C, [alpha]) summability method/Alguns teoremas Tauberiano estendidas para [(A).sup.(k)] (C, [alpha]) metodo de somabilidade.

Introduction

Let [summation][a.sub.n] be a given infinite series of real numbers with the sequence of n-th partial sums ([s.sub.n]) = ([[summation].sup.n.sub.k=0] [a.sub.k]). For a sequence ([s.sub.n]), we define [DELTA][s.sub.n] = [s.sub.n] - [s.sub.n-1], with [DELTA][s.sub.0] = [s.sub.0]. Let [A.sup.[alpha].sub.n] be defined by the generating function [(1 - x).sup.-[alpha]-1] = [[summation].sup.[infinity].sub.n=0] [A.sup.[alpha].sub.n] [x.sup.n] ([absolute value of (x)] < 1), where [alpha] > -1. A sequence ([s.sub.n]) is said to be (C, [alpha]) summable to S and we write [s.sub.n] [right arrow] s (C, [alpha]), if

[s.sup.[alpha].sub.n] = 1/[A.sup.[alpha].sub.n] [n.summation over (k=0)] [A.sup.[alpha]-1.sub.n-k] [s.sub.k] [right arrow] s

as n [right arrow] [infinity]. Note that (C,0) summability is the ordinary convergence. We write [[tau].sub.n] = [na.sub.n] and define [[tau].sup.[alpha].sub.n] as the (C, [alpha]) mean of [[tau].sub.n].

A sequence ([s.sub.n]) is said to be Abel summable to s, and we write [s.sub.n] [right arrow] s (A), if the series [[summation].sup.[infinity].sub.n=0][a.sub.n][x.sup.n] is convergent for 0 [less than or equal to] x < 1 and tends to s as x [right arrow] [1.sup.-]. A sequence ([s.sub.n]) is said to be (A) (C, [alpha]) summable to s and we write [s.sub.n] [right arrow] s (A) (C, [alpha]), if (1 - x)[[summation].sup.[infinity].sub.n=0][s.sup.[alpha].sub.n] [x.sup.n]: is convergent for 0 [less than or equal to] x < 1 and tends to s as x [right arrow] [1.sup.-]. If we take [alpha] = 0, then (A) (C, [alpha]) summability reduces to Abel summability.

A generalization of Abel summability is introduced by (LITTLEWOOD, 1967) as follows.

Let f (x) = [[summation].sup.[infinity].sub.n=0][a.sub.n][x.sup.n], 0 [less than or equal to] x < 1. Let

[f.sub.1](x) = 1/1 - x [[integral].sup.1.sub.x] f(t) dt,

and suppose that [[integral].sup.1.sub.0] f(t)dt exists as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let

[f.sub.2](x) = 1/1 - x f(t) dt,

an so on. We write

[f.sub.k](x) = 1/1 - x [[integral].sup.1x] [f.sub.k-1](t) dt

for positive integer k. The [f.sub.k] (x) is called the k-tuple average of f as x [right arrow] [1.sup.-] by (LITTLEWOOD, 1967). If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some positive integer k, we say that ([s.sub.n]) is [(A).sup.(k)] summable to s .

Let g(x) = (1 - x) [[summation].sup.[infinity].sub.n=0] [s.sup.[alpha].sub.n][x.sup.n], 0 [less than or equal to] x < 1, [alpha] > -1. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some positive integer k, we say that ([s.sub.n]) is [(A).sup.(k)](C, [alpha]) summable to s .

A sequence ([s.sub.n]) is said to be slowly oscillating (STANOJEVIC, 1998) if,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A sequence ([s.sub.n]) is said to be (C, [alpha]) slowly oscillating if ([s.sup.[alpha].sub.n]) is slowly oscillating.

We use the symbols [s.sub.n] = o(1), [s.sub.n] = O(1) to mean respectively that [s.sub.n] [right arrow] 0 as n [right arrow] [infinity] and that ([s.sub.n]) is bounded for large enough n. We also write [s.sub.n] = o(1) (C, [alpha]) to mean that [s.sup.[alpha].sub.n] = o(1).

Hardy (1910) proved that [na.sub.n] = O(1) is a Tauberian condition for (C, [alpha]), [alpha] > 0, summability of ([s.sub.n]). Later, Littlewood (1911) proved that (C, [alpha]) summability of ([s.sub.n]) in Hardy's theorem (HARDY, 1910) can be replaced by the Abel summability of ([s.sub.n]). (HARDY; LITTLEWOOD, 1913) replaced the condition n[a.sub.n] = O(1) by the one-sided Tauberian condition n[a.sub.n] [greater than or equal to] -H for some positive constant H.

Littlewood (1911) proved that if ([s.sub.n]) is Abel summable to S and [s.sub.n] = O(1), then ([s.sub.n]) is (C, 1) summable to S. Szasz (1935) generalized Littlewood's theorem (LITTLEWOOD, 1911) which states that if ([s.sub.n]) is Abel summable to s and [[tau].sup.1.sub.n] [greater than or equal to] -H for some positive constant H, then ([s.sub.n]) is (C, 1) summable to S. Pati (2002) obtained a more general theorem which states that if ([s.sub.n]) is (A) (C, [alpha]) summable for some [alpha] [greater than or equal to] 0 to s and [[tau].sup.[alpha].sub.n] [greater than or equal to] -H for some positive constant H, then ([s.sub.n]) is (C, [alpha]) summable to s. Quite recently, several new Tauberian conditions for (A) (C, [alpha]) summability method have been obtained in Canak et al. (2010), Erdem and Canak (2010), and Canak and Erdem, (2011).

Littlewood (1967) proved that n[a.sub.n] [greater than or equal to] -H for some positive constant H is a Tauberian condition for [(A).sup.(k)], where k is a positive integer k , summability of ([s.sub.n]). Pati (2007) established two Tauberian theorems which are more general than a theorem of Pati (2002) and a theorem of Littlewood (1967).

Our aim in this paper is to introduce some new conditions in terms of [[tau].sup.[alpha].sub.n] to recover (C, [alpha]) convergence of ([[tau].sub.n]) from its [(A).sup.(k)](C, [alpha]) summability. Namely, we prove the following Tauberian theorems.

Theorem 1.1

If, for some positive integer k and [alpha] [greater than or equal to] 0, ([[tau].sub.n]) is [(A).sup.(k)](C, [alpha]) summable to s and

n[DELTA][[tau].sup.[alpha].sub.n] = o(1) (1)

then ([[tau].sub.n]) is (C,[alpha] - 1) summable to s and ([s.sub.n]) is (C, [alpha] - 1) slowly oscillating.

Theorem 1.2

If, for some positive integer k and [alpha] [greater than or equal to] 0, ([[tau].sub.n]) is [(A).sup.(k)](C, [alpha]) summable to s and for some positive constant H

n[DELTA][[tau].sup.[alpha].sub.n] [greater than or equal to] -H (2)

then ([[tau].sub.n]) is (C, [alpha]) summable to s and ([s.sub.n]) is (C, [alpha]) slowly oscillating.

Theorem 1.3

If, for some positive integer k and [alpha] [greater than or equal to] 0, ([[tau].sub.n]) is [(A).sup.(k)](C, [alpha]) summable to s and for some positive constant H

n[DELTA][[tau].sup.[alpha].sub.n] = O(1) (3)

then ([[tau].sub.n]) is (C, [alpha] + [delta] - 1) summable to s for every [delta] > 0.

Proofs of our Theorems depend on the following Tauberian theorem due to Littlewood (1967).

Theorem 1.4

If for some positive integer k, ([s.sub.n]) is [(A).sup.(k)] summable to s, then n[a.sub.n] [greater than or equal to] -H for some positive constant H is a Tauberian condition for the convergence of ([s.sub.n]) to s .

Lemmas

For the proof of our theorems, we need the following lemmas.

Lemma 2.1

Kogbetliantz (1925, 1931) For [alpha] > -1, [[tau].sup.[alpha].sub.n] = n[DELTA] [s.sup.[alpha].sub.n] = n([s.sup.[alpha].sub.n] - [s.sup.[alpha].sub.n-1]).

Lemma 2.2

Canak et al. (2010) For

[alpha] > -1, n[DELTA] [[tau].sup.[alpha]+1.sub.n] = ([alpha] + 1)([[tau].sup.[alpha].sub.n] - [[tau].sup.[alpha]+1.sub.n]) (1)

Lemma 2.3

(HARDY, 1991) If [s.sup.[alpha].sub.n] [right arrow] s as n [right arrow] [infinity], [alpha] > -1, then [s.sup.[alpha]+[delta].sub.n] [right arrow] s as n [right arrow] [infinity] for every [delta] > 0.

Lemma 2.4

(HARDY, 1991) If [s.sup.[alpha].sub.n] [right arrow] s(C, [beta]), then [s.sup.[alpha]+[beta].sub.n] [right arrow] s for [alpha] [greater than or equal to] 0, [beta] [greater than or equal to] 0, and conversely.

Lemma 2.5

(PEYERIMHOFF, 1969) All the Cesaro methods of positive order are equivalent for bounded sequences. More precisely, if [s.sub.n] = 0(1) and [s.sup.[alpha].sub.n] [right arrow] s as n [right arrow] [infinity] for some [alpha] > 0, then [s.sup.[beta].sub.n] [right arrow] s as n [right arrow] [infinity] for some [beta] > 0 .

Proofs of Theorems

Proof of Theorem 1.1

By hypothesis, we have [f.sub.k] (x) [right arrow] s as x [right arrow] [1.sup.-], where [f.sub.k](x) is the k-tuple average of:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

The condition (1) implies that n[DELTA] [[tau].sup.[alpha].sub.n] [greater than or equal to] -H for some positive constant H . By Theorem 1.4, we get

[[infinity].summation over (n=0)] ([[tau].sup.[alpha].sub.n] - [[tau].sup.[alpha].sub.n-1]), ([[tau].sup.[alpha].sub.- 1] = 0) (5)

is convergent to s , i.e.,

[[tau].sup.[alpha].sub.n] [right arrow] s, n [right arrow] [infinity]. (6)

This means that ([[tau].sub.n]) is (C, [alpha]) summable to s .

By Lemma 2.2, we have

n[DELTA] [[tau].sup.[alpha].sub.n] = [alpha]([[tau].sup.[alpha]-1.sub.n] - [[tau].sup.[alpha].sub.n]). (7)

It follows from (1) and (6) that

[[tau].sup.[alpha]-1.sub.n] [right arrow] s, n [right arrow] [infinity], (8)

which means that ([[tau].sub.n]) is (C, [alpha] - 1) summable to s. By Lemma 2.1, we have

[s.sup.[alpha]-1.sub.n] = [n.summation over (k=1)] [[tau].sup.[alpha]-1.sub.k]/k. (9)

Since ([[tau].sup.[alpha]-1.sub.n]) converges to s, there exists M > 0 such that

[absolute value of ([[tau].sup.[alpha]-1.sub.n])] [less than or equal to] M (10)

for all n. For any n < k < [infinity], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

whence we conclude that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

Letting [lambda] [right arrow] [1.sup.+], we obtain ([s.sub.n]) is (C, [alpha] - 1) slowly oscillating. This completes the proof of Theorem 1.1.

Corollary 3.1

If, for some positive integer k, ([[tau].sub.n]) is [(A).sup.(k)] (C, 1) summable to s, and (1) holds, then ([[tau].sub.n]) is convergent to s and ([s.sub.n]) is slowly oscillating.

Proof

Take [alpha] = 1 in Theorem 1.1.

Proof of Theorem 1.2

We have ([[tau].sub.n]) is (C, [alpha]) summable to s by Theorem 1.4. That ([s.sub.n]) is (C, [alpha]) slowly oscillating follows from Lemma 2.2.

Proof of Theorem 1.3

The condition (3) implies that

n[DELTA] [[tau].sup.[alpha].sub.n] [greater than or equal to] -H (13)

for some positive constant H . By Theorem 1.2, we have

[[tau].sub.n] [right arrow] s(C, [alpha]). (14)

By Lemma 2.3,

[[tau].sub.n] [right arrow] s (C, [alpha] + 1) (15)

and by Lemma 2.2,

n[DELTA] [[tau].sup.[alpha]+1.sub.n] = [alpha]([[tau].sup.[alpha].sub.n] - [[tau].sup.[alpha]+1.sub.n]) = o(1), (16)

which is equivalent to

n[DELTA] [[tau].sup.[alpha].sub.n] = o(1)(C, 1) (17)

by Lemma 2.4. Since n[DELTA] [[tau].sup.[alpha].sub.n] = O(1) by hypothesis, we have, by Lemma 2.5,

n[DELTA] [[tau].sup.[alpha].sub.n] [right arrow] 0(C, [delta]) (18)

for every [delta] > 0, which is equivalent to

n[DELTA][[tau].sup.[alpha]+[delta].sub.n] = o(1) (19)

by Lemma 2.4.

By Lemma 2.2, we have

n[DELTA][[tau].sup.[alpha]+[delta].sub.n] = ([alpha] + [delta])([[tau].sup.[alpha]+[delta]-1.sub.n] - [[tau].sup.[alpha]+[delta].sub.n]) = o(1). (20)

By Lemma 2.3,

[[tau].sup.[alpha]+[delta].sub.n] [right arrow] s, n [right arrow] [infinity] (21)

It now follows from (20) that

[[tau].sup.[alpha]+[delta]-1.sub.n] [right arrow] s, n [right arrow] [infinity], (22)

which is equivalent to

[[tau].sub.n] [right arrow] s(C, [alpha] + [delta] - 1). (23)

This completes the proof of Theorem 1.3. Corollary 3.2

If, for some positive integer k, ([[tau].sub.n]) is [(A).sup.(k)] (C, 1) summable to S, and (3) holds, then ([[tau].sub.n]) is (C, [delta]) summable to S for every [delta] > 0.

Proof

Take [alpha] = 1 in Theorem 1.3.

Corollary 3.3

If, for some positive integer k and 0 < [alpha] < 1, ([[tau].sub.n]) is [(A).sup.(k)](C, [alpha]) summable to S, and (3) holds, then ([[tau].sub.n]) is convergent to S.

Proof

Take [delta] = 1 - [alpha] (0 < [alpha] < 1) in Theorem 1.3.

Corollary 3.4

If, for some positive integer k, ([[tau].sub.n]) is [(A).sup.(k)] summable to S, and

n[DELTA](n[a.sub.n]) = O(1), (24)

then ([[tau].sub.n]) is (C, [delta] - 1) summable to S for every [delta] > 0.

Proof

Take [alpha] = 0 in Theorem 1.3.

Conclusion

New Tauberian theorems for the product [(A).sup.(k)] and (C, [alpha]) summability methods have been established. Some new Tauberian conditions in terms of (C, [alpha]) mean of ([[tau].sub.n]) have been obtained to recover (C, [alpha]) convergence of ([[tau].sub.n]) and slow oscillation of (C, [alpha]) mean from [(A).sup.(k)](C, [alpha]) summability of ([[tau].sub.n]).

Doi: 10.4025/actascitechnol.v36i4.16605

Acknowledgements

The author thanks the referees for their comments on the paper.

References

CANAK, I.; ERDEM, Y. On Tauberian theorems for (A)(C,[alpha]) summability method. Applied Mathematics and Computation, v. 218, n. 6, p. 2829-2836, 2011.

CANAK, I.; ERDEM, Y.; TOTUR, U. Some Tauberian theorems for (A)(C ,n) summability method. Mathematical and Computer Modelling, v. 52, n. 5-6, p. 738-743, 2010.

ERDEM, Y.; CANAK, I. A Tauberian theorem for (A) (C, [alpha]) summability. Computers and Mathematics with Applications, v. 60, n. 11, p. 2920-2925, 2010.

HARDY, G. H. Theorems relating to the summability and convergence of slowly oscillating series. Proceedings of the London Mathematical Society, v. 8, n. 2, p. 301-320, 1910.

HARDY, G. H. Divergent Series. New York: Chelsea, 1991.

HARDY, G. H.; LITTLEWOOD, J. E. Tauberian theorems concerning power and Dirichlet's series whose coefficients are positive. Proceedings of the London Mathematical Society, v. 13, n. 2, p. 174-191, 1913.

KOGBETLIANTZ, E. Sur le series absolument sommables par la methode des moyennes arihtmetiques. Bulletin de la Societe Mathematique de France, v. 49, n. 2, p. 234-251, 1925.

KOGBETLIANTZ, E. Sommation des series et integrals divergentes par les moyennes arithmetiques et typiques. Memorial Science de Mathematique, v. 51, p. 1-84, 1931.

LITTLEWOOD, J. E. The converse of Abel's theorem on power series. Proceedings of the London Mathematical Society, v. 9, n. 2, p. 434-448, 1911.

LITTLEWOOD, J. E. A theorem about successive derivatives of a function and some Tauberian theorems. Journal of the London Mathematical Society, v. 42, n. 1, p. 169-179, 1967.

PATI, T. Extended Tauberian theorems. Proceeding National conference on recent developments in sequences, summability and fourier analysis. In:

RATH, D.; NANDA, S. (Ed.). New Delhi: Narosa Publishing House, 2002. p. 235-250.

PATI, T. An extension of Littlewood's "O" Tauberian theorem. Journal of the International Academy of Physical Sciences, v. 11, n. 1, p. 89-98, 2007.

PEYERIMHOFF, A. Lectures on Summability. Berlin: Springer-Verlag, 1969.

STANOJEVIC, C. V. Analysis of divergence: Control and management of divergent process, graduate research seminar lecture notes. In: CANAK, 1. (Ed.). Missouri: University of Missouri-Rolla, 1998. p. 1-56.

SZASZ, O. Generalization of two theorems of Hardy and Littlewood on power series. Duke Mathematical Journal, v. 1, n. 1, p. 105-111, 1935.

Accepted on July 10, 2013.

License information: This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Ibrahim Canak

Department of Mathematics, Ege University, 35100, Izmir, Turkey. E-mail: ibrahim.canak@ege.edu.tr