Summability methods which include the Riesz typical means. II

1971 ◽  
Vol 69 (2) ◽  
pp. 297-300 ◽  
Author(s):  
B. C. Russell
Keyword(s):  
Regular Sequence ◽  
Cesàro Summability ◽  
Convergence Factor ◽  
Summability Methods ◽  
Cesaro Summability ◽  
Factor Theorem ◽  
T Matrix ◽  
Image Position ◽  
Summability Matrix ◽  
Necessary And Sufficient

By making use of a convergence-factor theorem of Bosanquet(3), Cooke((4), Theorem I) gave conditions for a regular sequence-to-sequence summability matrix B to be at least as strong as Cesàro summability (C, κ) (κ > 0), namely:Theorem C. Let κ > 0. In order that the T-matrix B = (bρμ) shall satisfy B ⊇ (C, κ) it is necessary and sufficient thatIf 0 < κ ≤ 1 then (2) alone is necessary and sufficient.

Summability methods which include the Riesz typical means. I

1971 ◽  
Vol 69 (1) ◽  
pp. 99-106 ◽  
Author(s):  
Dennis C. Russell
Keyword(s):  
Complete Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Alternative Form ◽  
Cesàro Summability ◽  
Abel Summability ◽  
Summability Methods ◽  
Cesaro Summability ◽  
Riesz Summability ◽  
Necessary And Sufficient

A number of special results exist for summability methods B which, include Riesz summability (R,λ,k)—for example, when B is generalized Abel summability (A,λ,ρ) [Kuttner(5)], or Riemann summability (,λ,μ) [Russell(14)], or Riemann-Cesàro summability (,λ,p,α) [Rangachari(12)], or generalized Cesàro summability (C,λ,k) [Meir (9); Borwein and Russell (l)]. The question of necessary and sufficient conditions to be satisfied by an arbitrary method B in order that B ⊇ (R,λ,k) has received an answer only for limited values of λ and k—for example, by Lorentz [(6), Theorem 10] for k = 1; the restrictions on λ in this case were removed by Maddox [(8), Theorem 1]. Thus (apart from the well-known case k = 0) the case k = 1 is the only one for which a complete solution exists, though application of a theorem of Russell [(13), Theorem 1A] yields one form of a result for 0 < k ≤ 1. Maddox's results, however, suggest an alternative form capable of generalization to all k ≥ 0, and in this paper we obtain a complete solution for 0 < k ≤ 1 in that form, without restriction on λ. We first recall the following definitions.

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.

scholarly journals Tauberian Theorems for the Product of Weighted and Cesàro Summability Methods for Double Sequences

2019 ◽  
Vol 38 (7) ◽  
pp. 9-19
Author(s):  
Gökşen Fındık ◽  
İbrahim Çanak
Keyword(s):  
Double Sequence ◽  
Sufficient Conditions ◽  
Complex Numbers ◽  
Tauberian Theorems ◽  
Double Sequences ◽  
Cesàro Summability ◽  
Tauberian Conditions ◽  
Summability Methods ◽  
Cesaro Summability ◽  
Necessary And Sufficient

In this paper, we obtain necessary and sufficient conditions, under which convergence of a double sequence in Pringsheim's sense follows from its weighted-Cesaro summability. These Tauberian conditions are one-sided or two-sided if it is a sequence of real or complex numbers, respectively.

scholarly journals Summability methods based on the Riemann Zeta function

1988 ◽  
Vol 11 (1) ◽  
pp. 27-36
Author(s):  
Larry K. Chu
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Necessary Condition ◽  
Real Sequence ◽  
Necessary And Sufficient Condition ◽  
Summability Methods ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Summability Matrix ◽  
Necessary And Sufficient

This paper is a study of summability methods that are based on the Riemann Zeta function. A limitation theorem is proved which gives a necessary condition for a sequencexto be zeta summable. A zeta summability matrixZtassociated with a real sequencetis introduced; a necessary and sufficient condition on the sequencetsuch thatZtmapsl1tol1is established. Results comparing the strength of the zeta method to that of well-known summability methods are also investigated.

scholarly journals The Absolute Cesaro Summability of the Successively Derived Allied Series of a Fourier Series

1950 ◽  
Vol 8 (4) ◽  
pp. 163-176
Author(s):  
R. Mohanty
Keyword(s):  
Fourier Series ◽  
Cesàro Summability ◽  
Cesaro Summability ◽  
The Absolute ◽  
Image Position

We suppose that f(t) is integrable in the Lebesgue sense m (π, π) and is periodic with period 2π. We denote its Fourier series byThen the allied series is

scholarly journals A Definition for Strong Rieszian Summability and its Relationship to Strong Cesaro Summability

1952 ◽  
Vol 1 (2) ◽  
pp. 94-99 ◽  
Author(s):  
A. V. Boyd ◽  
J. M. Hyslop
Keyword(s):  
Cesàro Summability ◽  
Cesaro Summability ◽  
Image Position

1. Introduction. Given a series we define , by the relationsThe series Σan is said to be summable (C, k) to the sum s, ifas n→∞, and strongly summable (C, k), k>0, with index p, to the sum s, or summable [C; k, p] to the sum s, if

scholarly journals Some sufficient conditions for the absolute Cesàro summability of series

1939 ◽  
Vol 6 (1) ◽  
pp. 51-56 ◽  
Author(s):  
J. M. Hyslop
Keyword(s):  
Sufficient Conditions ◽  
Cesàro Summability ◽  
Cesàro Mean ◽  
Cesaro Summability ◽  
Cesaro Mean ◽  
The Absolute ◽  
Image Position

denote the n-th Cesàro mean of order k for the series aΣan, that is,whereand let

scholarly journals Some theorems on absolute Cesàro summability

1939 ◽  
Vol 6 (2) ◽  
pp. 114-122 ◽  
Author(s):  
J. M. Hyslop
Keyword(s):  
Cesàro Summability ◽  
Cesaro Summability ◽  
Formal Expansion ◽  
Image Position

1. It is convenient to begin with a brief statement of the notation which will be used throughout this paper.Let k be any positive number and letwhere is the coefficient of xn in the formal expansion of (1 – x )–k–1, and letThen the series Σαn is said to be summable(C, k) if is convergent, that is, if tends to a limit, and absolutely summable (C, k), or summable |C, k|, if is absolutely convergent.

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