scholarly journals Equivalence results for discrete Abel means

2002 ◽  
Vol 30 (12) ◽  
pp. 727-731
Author(s):  
Jeffrey A. Osikiewicz
Keyword(s):  
Summability Method ◽  
Abel Summability ◽  
Abel Means ◽  
Abel Mean ◽  
Bounded Sequences

We present theorems showing when the discrete Abel mean and the Abel summability method are equivalent for bounded sequences and when two discrete Abel means are equivalent for bounded sequences.

scholarly journals A Tauberian Theorem for a General Summability Method

2015 ◽  
Vol 61 (1) ◽  
pp. 123-128
Author(s):  
Ibrahim Çanak
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Abel Summability ◽  
General Summability

Abstract We investigate conditions under which Mϕ summability implies Abel summability and give the generalized Littlewood Tauberian theorem for Mϕ summability method.

Abel extensions of some classical Tauberian theorems

2019 ◽  
Vol 28 (2) ◽  
pp. 105-112
Author(s):  
ERDAL GUL ◽  
MEHMET ALBAYRAK
Keyword(s):  
Tauberian Theorem ◽  
Matrix Method ◽  
Tauberian Theorems ◽  
Borel Summability ◽  
Tauberian Condition ◽  
Continuous Parameter ◽  
Abel Means ◽  
The One ◽  
Abel Mean

The well-known classical Tauberian theorems given for Aλ (the discrete Abel mean) by Armitage and Maddox in [Armitage, H. D and Maddox, J. I., Discrete Abel means, Analysis, 10 (1990), 177–186] is generalized. Similarly the ”one-sided” Tauberian theorems of Landau and Schmidt for the Abel method are extended by replacing lim As with Abel-lim Aσi n(s). Slowly oscillating of {sn} is a Tauberian condition of the Hardy-Littlewood Tauberian theorem for Borel summability which is also given by replacing limt(Bs)t = `, where t is a continuous parameter, with limn(Bs)n = `, and further replacing it by Abel-lim(Bσi k (s))n = `, where B is the Borel matrix method.

On ‘translated quasi-Cesàro’ summability. II

1969 ◽  
Vol 66 (1) ◽  
pp. 39-41 ◽  
Author(s):  
B. Kuttner
Keyword(s):  
Summability Method ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Cesàro Summability ◽  
Abel Summability ◽  
Cesaro Summability ◽  
Sequence Transformation ◽  
Image Position ◽  
Necessary And Sufficient

In a recent paper (1), I considered the summability method (D, α) defined, for α > 0, by the sequence-to-sequence transformationWe note that, as is easily verified (and as was pointed out in (1)) a necessary and sufficient condition for the convergence of (1), and thus for the applicability of (D, α), is thatshould converge. It was proved in (1) that, provided that (2) converges, a sequence summable (C, r) for any r > − 1 is necessarily summable (D, α). We now show that we can strengthen this result by replacing Cesàro by Abel summability. Moreover, we can omit the hypothesis that (2) converges provided that we interpret (1) as an Abel sum.

On Strong Matrix Summability with Respect to a Modulus and Statistical Convergence

1989 ◽  
Vol 32 (2) ◽  
pp. 194-198 ◽  
Author(s):  
Jeff Connor
Keyword(s):  
Statistical Convergence ◽  
Summability Method ◽  
Regular Matrix ◽  
Cesàro Summability ◽  
Matrix Summability ◽  
Cesaro Summability ◽  
Definition Of ◽  
Bounded Sequences

AbstractThe definition of strong Cesaro summability with respect to a modulus is extended to a definition of strong A -summability with respect to a modulus when A is a nonnegative regular matrix summability method. It is shown that if a sequence is strongly A-summable with respect to an arbitrary modulus then it is A-statistically convergent and that Astatistical convergence and strong A-summability with respect to a modulus are equivalent on the bounded sequences.

Abel Transformations Into I1

1982 ◽  
Vol 25 (4) ◽  
pp. 421-427 ◽  
Author(s):  
J. A. Fridy
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Partial Sums ◽  
Sequence Variant ◽  
Abel Summability ◽  
Summability Methods ◽  
The Matrix ◽  
Bounded Partial Sums

AbstractLet t be a sequence in (0,1) that converges to 0, and define the Abel matrix At by ank = tn(1-tn)k. The matrix At determines a sequence-to-sequence variant of the classical Abel summability method. The purpose of this paper is to study these transformations as l-l summability methods: e.g., At maps l1 into l1 if and only if t is in l1. The Abel matrices are shown to be stronger l-l methods than the Euler-Knopp means and the Nӧrlund means. Indeed, if t is in l1 and Σ xk has bounded partial sums, then Atx is in l1. Also, the Abel matrix is shown to be translative in an l-l sense, and an l-l Tauberian theorem is proved for At.

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

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Ümit Totur
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Subject Classification ◽  
Tauberian Theorems ◽  
Double Sequences ◽  
Littlewood Theorem

Abstract In this paper we generalize some classical Tauberian theorems for single sequences to double sequences. One-sided Tauberian theorem and generalized Littlewood theorem for (C; 1; 1) summability method are given as corollaries of the main results. Mathematics Subject Classification 2010: 40E05, 40G0

scholarly journals Approximation by a power series summability method of Kantorovich type Szász operators including Sheffer polynomials

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Valdete Loku ◽  
Naim L. Braha ◽  
Toufik Mansour ◽  
M. Mursaleen
Keyword(s):  
Power Series ◽  
Summability Method ◽  
Approximation Properties ◽  
Type Result ◽  
Szász Operators ◽  
Sheffer Polynomials

AbstractThe main purpose of this paper is to use a power series summability method to study some approximation properties of Kantorovich type Szász–Mirakyan operators including Sheffer polynomials. We also establish Voronovskaya type result.

Orlicz lacunary sequence spaces of 𝑙-fractional difference operators

2020 ◽  
Vol 26 (2) ◽  
pp. 173-183
Author(s):  
Kuldip Raj ◽  
Kavita Saini ◽  
Anu Choudhary
Keyword(s):  
Difference Operator ◽  
Lacunary Sequence ◽  
Sequence Spaces ◽  
Difference Operators ◽  
Normed Spaces ◽  
Fractional Difference ◽  
New Approach ◽  
Inclusion Relations ◽  
Difference Sequence Spaces ◽  
Bounded Sequences

AbstractRecently, S. K. Mahato and P. D. Srivastava [A class of sequence spaces defined by 𝑙-fractional difference operator, preprint 2018, http://arxiv.org/abs/1806.10383] studied 𝑙-fractional difference sequence spaces. In this article, we intend to make a new approach to introduce and study some lambda 𝑙-fractional convergent, lambda 𝑙-fractional null and lambda 𝑙-fractional bounded sequences over 𝑛-normed spaces. Various algebraic and topological properties of these newly formed sequence spaces have been explored, and some inclusion relations concerning these spaces are also established. Finally, some characterizations of the newly formed sequence spaces are given.

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