scholarly journals A revisited Tauberian theorem for which slow decrease with respect to a weight function is a tauberian condition for the weighted mean summability of integrals over R+

2021 ◽  
Author(s):  
Ibrahim Canak
Keyword(s):  
Weight Function ◽  
Tauberian Theorem ◽  
Weighted Mean ◽  
Tauberian Condition ◽  
Slow Decrease

scholarly journals Gap Tauberian theorems

1993 ◽  
Vol 47 (3) ◽  
pp. 385-393 ◽  
Author(s):  
Jeff Connor
Keyword(s):  
Tauberian Theorem ◽  
Strong Summability ◽  
Tauberian Theorems ◽  
Weighted Mean ◽  
Tauberian Condition ◽  
Support Set ◽  
Tauberian Conditions ◽  
Summability Matrix ◽  
Regular Summability ◽  
Bk Spaces

In the first section we establish a connection between gap Tauberian conditions and isomorphic copies of Co for perfect coregular conservative BK spaces and in the second we give a characterisation of gap Tauberian conditions for strong summability with respect to a nonnnegative regular summability matrix. These results are used to show that a gap Tauberian condition for strong weighted mean summability is also a gap Tauberian condition for ordinary weighted mean summability. We also make a remark regarding the support set of a matrix and give a Tauberian theorem for a class of conull spaces.

scholarly journals GENERALIZATION OF THE EFFECTIVE WIENER–IKEHARA THEOREM

2013 ◽  
Vol 09 (08) ◽  
pp. 2091-2128 ◽  
Author(s):  
SZILÁRD GY. RÉVÉSZ ◽  
ANNE de ROTON
Keyword(s):  
Laplace Transform ◽  
Tauberian Theorem ◽  
Error Term ◽  
Effective Version ◽  
Slow Decrease ◽  
Asymptotic Evaluation ◽  
The Laplace Transform

We consider the classical Wiener–Ikehara Tauberian theorem, with a generalized condition of slow decrease and some additional poles on the boundary of convergence of the Laplace transform. In this generality, we prove the otherwise known asymptotic evaluation of the transformed function, when the usual conditions of the Wiener–Ikehara theorem hold. However, our version also provides an effective error term, not known thus far in this generality. The crux of the proof is a proper, asymptotic variation of the lemmas of Ganelius and Tenenbaum, also constructed for the sake of an effective version of the Wiener–Ikehara theorem.

A Tauberian Theorem for the General Euler-Borel Summability Method

1992 ◽  
Vol 44 (5) ◽  
pp. 1100-1120 ◽  
Author(s):  
Laying Tam
Keyword(s):  
Positive Integer ◽  
Unit Disk ◽  
Tauberian Theorem ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Borel Summability ◽  
Closed Unit Disk ◽  
Tauberian Condition ◽  
Image Position ◽  
General Conditions

AbstractOur main result is a Tauberian theorem for the general Euler-Borel summability method. Examples of the method include the discrete Borel, Euler, Meyer- Kônig, Taylor and Karamata methods. Every function/ analytic on the closed unit disk and satisfying some general conditions generates such a method, denoted by (£,ƒ). For instance the function ƒ(z) = exp(z — 1) generates the discrete Borel method. To each such function ƒ corresponds an even positive integer p = p(f).We show that if a sequence (sn)is summable (E,f)and as n→ ∞ m > n, (m— n)n-p(f)→0, then (sn)is convergent. If the Maclaurin coefficients of/ are nonnegative, then p(f) =2. In this case we may replace the condition . This generalizes the Tauberian theorems for Borel summability due to Hardy and Littlewood, and R. Schmidt. We have also found new examples of the method and proved that the exponent —p(f)in the Tauberian condition (*) is the best possible.

A Tauberian theorem for discrete weighted mean methods

Analysis ◽  
2006 ◽  
Vol 26 (4) ◽  
Author(s):  
Bruce Watson
Keyword(s):  
Tauberian Theorem ◽  
Weighted Mean

A Tauberian theorem of “slowly decreasing” type is proved for discrete weighted mean methods of summability by reduction to the corresponding Tauberian theorem for weighted mean methods.

A Tauberian theorem for statistical convergence

1989 ◽  
Vol 106 (2) ◽  
pp. 277-280 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Linear Space ◽  
Tauberian Theorem ◽  
Statistical Convergence ◽  
Slow Oscillation ◽  
Topological Linear Space ◽  
Tauberian Condition ◽  
Topological Linear ◽  
Locally Convex

The notion of statistical convergence of a sequence (xk) in a locally convex Hausdorff topological linear space X was introduced recently by Maddox[5], where it was shown that the slow oscillation of (sk) was a Tauberian condition for the statistical convergence of (sk).

A note on cosine series with coefficients of generalized bounded variation

2020 ◽  
Vol 70 (3) ◽  
pp. 681-688
Author(s):  
Bhikha Lila Ghodadra ◽  
Vanda Fülöp
Keyword(s):  
Bounded Variation ◽  
Tauberian Theorem ◽  
Tauberian Theorems ◽  
Weighted Mean ◽  
Triangular Matrices ◽  
Cosine Series ◽  
Lower Triangular ◽  
Hausdorff Matrices ◽  
Lower Triangular Matrices

AbstractIn this note, we obtain a Tauberian theorem for a class of regular lower triangular matrices operating on cosine series with coefficients tending to zero. As corollaries we obtain Tauberian theorems for weighted mean, Nörlund, and Hausdorff matrices.

scholarly journals TAUBERIAN THEOREMS FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY OF INTEGRALS

2020 ◽  
pp. 775
Author(s):  
Firat Ozsarac ◽  
Ibrahim Canak
Keyword(s):  
Finite Number ◽  
Weight Function ◽  
Generating Function ◽  
Finite Interval ◽  
Classical Type ◽  
Tauberian Theorems ◽  
Weighted Mean ◽  
Positive Weight ◽  
Tauberian Conditions ◽  
Complex Valued

Let $q$ be a positive weight function on $\mathbf{R}_{+}:=[0, \infty)$ which is integrable in Lebesgue's sense over every finite interval $(0,x)$ for $0<x<\infty$, in symbol: $q \in L^{1}_{loc} (\mathbf{R}_{+})$ such that $Q(x):=\int_{0}^{x} q(t) dt\neq 0$ for each $x>0$, $Q(0)=0$ and $Q(x) \rightarrow \infty $ as $x \to \infty $.Given a real or complex-valued function $f \in L^{1}_{loc} (\mathbf{R}_{+})$, we define $s(x):=\int_{0}^{x}f(t)dt$ and$$\tau^{(0)}_q(x):=s(x), \tau^{(m)}_q(x):=\frac{1}{Q(x)}\int_0^x \tau^{(m-1)}_q(t) q(t)dt\,\,\, (x>0, m=1,2,...),$$provided that $Q(x)>0$. We say that $\int_{0}^{\infty}f(x)dx$ is summable to $L$ by the $m$-th iteration of weighted mean method determined by the function $q(x)$, or for short, $(\overline{N},q,m)$ integrable to a finite number $L$ if$$\lim_{x\to \infty}\tau^{(m)}_q(x)=L.$$In this case, we write $s(x)\rightarrow L(\overline{N},q,m)$. It is known thatif the limit $\lim _{x \to \infty} s(x)=L$ exists, then $\lim _{x \to \infty} \tau^{(m)}_q(x)=L$ also exists. However, the converse of this implicationis not always true. Some suitable conditions together with the existence of the limit $\lim _{x \to \infty} \tau^{(m)}_q(x)$, which is so called Tauberian conditions, may imply convergence of $\lim _{x \to \infty} s(x)$. In this paper, one- and two-sided Tauberian conditions in terms of the generating function and its generalizations for $(\overline{N},q,m)$ summable integrals of real- or complex-valued functions have been obtained. Some classical type Tauberian theorems given for Ces\`{a}ro summability $(C,1)$ and weighted mean method of summability $(\overline{N},q)$ have been extended and generalized.  

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.

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