Absolute summability of infinite series on a scale of Abel type summability methods

1968 ◽  
Vol 64 (2) ◽  
pp. 377-387 ◽  
Author(s):  
Babban Prasad Mishra
Keyword(s):  
Infinite Series ◽  
Open Interval ◽  
Absolute Summability ◽  
Finite Limit ◽  
Summability Methods ◽  
Image Position

Suppose that λ > − 1 and thatIt is easy to show thatWith Borwein(1), we say that the sequence {sn} is summable Aλ to s, and write sn → s(Aλ), if the seriesis convergent for all x in the open interval (0, 1)and tends to a finite limit s as x → 1 in (0, 1). The A0 method is the ordinary Abel method.

Strong summability of infinite series on a scale of Abel type summability methods

1967 ◽  
Vol 63 (1) ◽  
pp. 119-127 ◽  
Author(s):  
Babban Prasad Mishra
Keyword(s):  
Infinite Series ◽  
Open Interval ◽  
Strong Summability ◽  
New Method ◽  
Finite Limit ◽  
Summability Methods ◽  
Image Position

Introduction. In a recent paper, Borwein(1) constructed a new method of summability which would read: Letand let {sn} be any sequence of numbers. If, for λ > − 1,is convergent for all x in the open interval (0,1) and tends to a finite limit s as x → 1 in (0,1), we say that the sequence {sn} is Aλ convergent to s and write sn → s(Aλ). The A0 method is the ordinary Abel method.

On a scale of Abel-type summability methods

1957 ◽  
Vol 53 (2) ◽  
pp. 318-322 ◽  
Author(s):  
D. Borwein
Keyword(s):  
Open Interval ◽  
Finite Limit ◽  
Summability Methods ◽  
Image Position

1. Introduction. In this note Abel-type summability methods (Aλ) are defined and some of their properties investigated.Letand let {sn} be any sequence of numbers. Ifis convergent for all x in the open interval (0,1) and tends to a finite limit s as x → 1 in (0,1), we shall say that the sequence is Aλ-convergent to s and write sn → s (Aλ;). The A0 method is the ordinary Abel method.

On the summability of Fourier integrals

1970 ◽  
Vol 67 (1) ◽  
pp. 23-28
Author(s):  
M. K. Nayak
Keyword(s):  
Open Interval ◽  
Finite Limit ◽  
Image Position ◽  
Fourier Integrals

We say a series is summable L iftends to a finite limit s as x → 1 in the open interval (0, 1) where

On Absolute Summability by Riesz and Generalized Cesàro Means. I

1970 ◽  
Vol 22 (2) ◽  
pp. 202-208 ◽  
Author(s):  
H.-H. Körle
Keyword(s):  
Infinite Series ◽  
Absolute Summability ◽  
Partial Sums ◽  
Cesàro Means ◽  
Riesz Means ◽  
Cesaro Means ◽  
Cesáro Means ◽  
Image Position ◽  
Generalized Cesàro Means

1. The Cesàro methods for ordinary [9, p. 17; 6, p. 96] and for absolute [9, p. 25] summation of infinite series can be generalized by the Riesz methods [7, p. 21; 12; 9, p. 52; 6, p. 86; 5, p. 2] and by “the generalized Cesàro methods“ introduced by Burkill [4] and Borwein and Russell [3]. (Also cf. [2]; for another generalization, see [8].) These generalizations raise the question as to their equivalence.We shall consider series(1)with complex terms an. Throughout, we will assume that(2)and we call (1) Riesz summable to a sum s relative to the type λ = (λn) and to the order κ, or summable (R, λ, κ) to s briefly, if the Riesz means(of the partial sums of (1)) tend to s as x → ∞.

scholarly journals On the Absolute Summability (A) of Infinite Series

1932 ◽  
Vol 3 (2) ◽  
pp. 132-134 ◽  
Author(s):  
M Fekete
Keyword(s):  
Bounded Variation ◽  
Infinite Series ◽  
Absolute Summability ◽  
The Absolute ◽  
Image Position

§1. A serieshas been defined by J. M. Whittaker to be absolutely summable (A), ifis convergent in (0 ≤ x < 1) and f (x) is of bounded variation in (0, 1), i.e.for all subdivisions 0 = x0 < x1 < x2 < . … < xm < 1.

On absolute summability factors of infinite series and their application to Fourier series

1967 ◽  
Vol 63 (1) ◽  
pp. 107-118 ◽  
Author(s):  
R. N. Mohapatra ◽  
G. Das ◽  
V. P. Srivastava
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Absolute Summability ◽  
Summability Factors ◽  
Partial Sum ◽  
Absolute Summability Factors ◽  
Image Position

Definition. Let {sn} be the n-th partial sum of a given infinite series. If the transformationwhereis a sequence of bounded variation, we say that εanis summable |C, α|.

A Tauberian theorem for absolute summability

1970 ◽  
Vol 67 (2) ◽  
pp. 321-326 ◽  
Author(s):  
G. Das
Keyword(s):  
Power Series ◽  
Infinite Series ◽  
Tauberian Theorem ◽  
Power Series Expansion ◽  
Binomial Coefficient ◽  
Absolute Summability ◽  
Partial Sums ◽  
Function Of Bounded Variation ◽  
Image Position ◽  
The Given

Let be the given infinite series with {sn} as the sequence of partial sums and let be the binomial coefficient of zn in the power series expansion of the function (l-z)-σ-1 |z| < 1. Now let, for β > – 1,converge for 0 ≤ x < 1. If fβ(x) → s as x → 1–, then we say that ∑an is summable (Aβ) to s. If, further, f(x) is a function of bounded variation in (0, 1), then ∑an is summable |Aβ| or absolutely summable (Aβ). We write this symbolically as {sn} ∈ |Aβ|. This method was first introduced by Borwein in (l) where he proves that for α > β > -1, (Aα) ⊂ (Aβ). Note that for β = 0, (Aβ) is the same as Abel method (A). Borwein (2) also introduced the (C, α, β) method as follows: Let α + β ╪ −1, −2, … Then the (C, α, β) mean is defined by

On two summability methods

1985 ◽  
Vol 97 (1) ◽  
pp. 147-149 ◽  
Author(s):  
Hüseyin Bor
Keyword(s):  
Infinite Series ◽  
Partial Sums ◽  
Cesàro Means ◽  
Cesaro Means ◽  
Summability Methods ◽  
Cesáro Means ◽  
Image Position

Let Σan be a given infinite series with partial sums sn, and rn = nan. By and we denote the nth Cesáro means of order α (α –1) of the sequences (sn) and (rn), respectively. The series Σan is said to be absolutely summable (C, a) with index k, or simply summable |C, α|k, k ≥ 1, if

XVII.—Relations between the Elliptic Cylinder Functions

1940 ◽  
Vol 59 ◽  
pp. 176-183 ◽  
Author(s):  
E. L. Ince
Keyword(s):  
Infinite Series ◽  
London Math ◽  
Recurrence Relations ◽  
Mathieu Equation ◽  
Elliptic Cylinder ◽  
Open Interval ◽  
Distinct Species ◽  
Number Of Zeros ◽  
Image Position ◽  
The Right

The elliptic cylinder functions, or solutions of the Mathieu equationthat have the period π or 2π, are of four distinct species. These are denoted respectively by ce2n(x, θ), se2n+1 (x, θ), ce2n+1 (x, θ), se2n+2(x, θ), where n indicates the number of zeros in the open interval o < x < ½π. The object of this paper is to show that a function of any type can be expressed in terms of functions of other types, or of their derivatives. This provides the analogies to such trigonometrical relations asbut the single terms on the right-hand side of this and similar relations are replaced, in the results to be proved, by infinite series. These results are quite distinct from Whittaker's recurrence-relations (Journ. London Math. Soc., vol. iv, 1929, pp. 88–96); in particular, they do not involve the non-periodic second solutions.

Quotient groups and realization of tight Riesz groups

1973 ◽  
Vol 16 (4) ◽  
pp. 416-430 ◽  
Author(s):  
John Boris Miller
Keyword(s):  
Normal Subgroup ◽  
Open Interval ◽  
Canonical Homomorphism ◽  
Interval Topology ◽  
Essential Step ◽  
Image Position ◽  
Quotient Groups

Let (G, ≼) be an l-group having a compatible tight Riesz order ≦ with open-interval topology U, and H a normal subgroup. The first part of the paper concerns the question: Under what conditions on H is the structure of (G, ≼, ∧, ∨, ≦, U) carried over satisfactorily to by the canonical homomorphism; and its answer (Theorem 8°): H should be an l-ideal of (G, ≼) closed and not open in (G, U). Such a normal subgroup is here called a tangent. An essential step is to show that ≼′ is the associated order of ≦′.

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