A Tauberian Theorem for Borel-Type Methods of Summability

1969 ◽  
Vol 21 ◽  
pp. 740-747 ◽  
Author(s):  
D. Borwein
Keyword(s):  
Tauberian Theorem ◽  
Convergent Series ◽  
Summability Method ◽  
Negative Integer ◽  
Exponential Method ◽  
Image Position ◽  
Borel Type ◽  
Cesàro Method

Suppose throughout that α >0, β is real, and Nis a non-negative integer such that αN+ β> 0. A series of complex terms is said to be summable (B, α,β) to l if, as x→ ∞,where sn= a0 + a1 + … + an.The Borel-type summability method (B, α, β) is regular, i.e., all convergent series are summable (B, α,β) to their natural sums; and (B,1, 1) is the standard Borel exponential method B.Our aim in this paper is to prove the following Tauberian theorem.THEOREM. Iƒ(i) p ≧ – ½, an = o(np), and(ii) is summable (B, α,β) to l, then the series is summable by the Cesaro method(C, 2p + 1) to l.

scholarly journals A Tauberian Theorem Concerning Borel-Type and Cesàro Methods of Summability

1988 ◽  
Vol 40 (1) ◽  
pp. 228-247 ◽  
Author(s):  
David Borwein ◽  
Tom Markovich
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Negative Integer ◽  
Real Sequence ◽  
Real Numbers ◽  
Tauberian Condition ◽  
Cesaro Summability ◽  
Exponential Method ◽  
Image Position ◽  
Borel Type

Suppose throughout that r ≧ 0, α > 0, αq + β > 0 where q is a non-negative integer. Let {sn} be a sequence of real numbers,The Borel-type summability method (B, α, β) is defined as follows:The method (B, α, β) is regular [5]; and (B, 1, 1) is the standard Borel exponential method B. For a real sequence {sn} we consider the slowly decreasing-type Tauberian conditionWe shall also be concerned with the Cesàro summability method Cp(p > —1), the Valiron method Vα, and the Meyer-König method Sa (0 < a < 1) defined as follows:

Absolute summability functions for Hausdorff methods

1982 ◽  
Vol 91 (1) ◽  
pp. 51-56
Author(s):  
B. Kuttner
Keyword(s):  
Absolute Summability ◽  
Bounded Sequence ◽  
Summability Method ◽  
Negative Integer ◽  
Hausdorff Method ◽  
The Matrix ◽  
Sequence Transformation ◽  
Image Position

1. Following Lorentz, we suppose throughout that Ω(n) is a non-negati ve non-decreasing function of the non-negative integer n such that Ω(n)→ ∞ as n → ∞. Consider the summability method given by the sequence-to-sequence transformation corresponding to the matrix A = (ank). We say that Ω(n) is a summability function for A (or absolute summability function for A) if the following holds: Any bounded sequence {sn} such that the number of values of ν with ν ≤ n, sν ≠ 0 does not exceed Ω(n) is summable A (or is absolutely summable A, respectively). These definitions are due to Lorentz (4), (6). We shall be concerned with the case in which A is a regular Hausdorff method, say A = H = (H, μn). Then H is given by the matrix (hnk) withwithX(0) = X(0 + ) = 0, X(1) = 1;(see e.g.(1), chapter XI). We shall suppose throughout that these conditions are satisfied. It is known that H is then necessarily also absolutely regular.

A Tauberian Theorem for a Scale of Logarithmic Methods of Summation

1973 ◽  
Vol 25 (5) ◽  
pp. 897-902 ◽  
Author(s):  
R. Phillips
Keyword(s):  
Tauberian Theorem ◽  
Negative Integer ◽  
Image Position

We suppose throughout that p is a non-negative integer, and use the following notations:where (n = 0 , 1 , 2 , … );

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.

Tauberian Theorems for Borel-Type Methods of Summability

1974 ◽  
Vol 17 (2) ◽  
pp. 167-173 ◽  
Author(s):  
D. Borwein ◽  
E. Smet
Keyword(s):  
Complex Variable ◽  
Negative Integer ◽  
Complex Numbers ◽  
Tauberian Theorems ◽  
Arbitrary Complex ◽  
Principal Value ◽  
Image Position ◽  
Borel Type

Suppose throughout that s, an (n=0,1, 2,…) are arbitrary complex numbers, that α>0 and β is real and that N is a non-negative integer such that αN+β≧l. Letwhere z=x+iy is a complex variable and the power zr is assumed to have its principal value.

scholarly journals Tauberian Theorems for Strong and Absolute Borel-Type Methods of Summability(1)

1977 ◽  
Vol 20 (2) ◽  
pp. 161-172
Author(s):  
D. Borwein ◽  
E. Smet
Keyword(s):  
Negative Integer ◽  
Complex Numbers ◽  
Tauberian Theorems ◽  
Arbitrary Complex ◽  
Image Position ◽  
Borel Type

Suppose throughout that s, an (n = 0,1,2,…) are arbitrary complex numbers, that α > 0 and β is real and that N is a non-negative integer such that αN + β≥1. Let

A Tauberian Theorem Concerning Borel-Type and Riesz Summability Methods

1992 ◽  
Vol 35 (1) ◽  
pp. 14-20 ◽  
Author(s):  
David Borwein
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Tauberian Condition ◽  
Summability Methods ◽  
Riesz Method ◽  
Riesz Summability ◽  
Borel Type

AbstractIt is proved that the summability of a series by the Borel-type summability method (B,α,β) together with a certain Tauberian condition implies its summability by the Riesz method (R, log(n + l),p).

On the Relation between the Logarithmic and Borel-Type Summability Methods

1981 ◽  
Vol 24 (2) ◽  
pp. 153-159
Author(s):  
D. Borwein ◽  
B. Watson
Keyword(s):  
Summability Method ◽  
Negative Integer ◽  
Real Numbers ◽  
Type Method ◽  
Summability Methods ◽  
Logarithmic Summability ◽  
Borel Type

Suppose throughout that {sn} is a sequence of real numbers, λ > - 1, a > 0, and β is real. Let N be any non-negative integer such that αN + β > l.We are concerned primarily with the logarithmic summability method L and the Borel-type method (B, α, β). Some known results involve the Abel-type summability method Aγ.

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

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