Abel Method Research Articles

Localization in bounded summation by Cesaro, Riesz, and Abel methods for multiple Fourier series

We establish the conditions under which the localization principle is valid in bounded summation by Cesaro, Riesz, and Abel methods for multiple Fourier series of functions of Liouville classes.

Uniform summability of power series

AbstractIt is shown that for the Cesàro means (C, α) with α > - 1, and for a certain class of more general Nörlund means, summability of the series σan implies uniform summability of the series σan zn in a Stolz angle at z = 1.If B is a normal matrix and (B) denotes the series summability field with the usual Banach space topology, then the vectors {ek} (ek = {0,0,..., 1,0,...}) are said to form a Toplitz basis for (B) relative to a method H if H — Σakek = a for each a = {ak}ε(B). It is shown for example that the above relation holds for B = (C,α), α> − 1 , and H = Abel method; also for B = (C,α) and H = (C,β) with 0 ≤ α ≤ β.Applications are made to theorems on summability factors.

On the Relation between the Abel-Type and Borel-Type Methods of Summability

In recent papers, R. Jajte and B. L. R. Shawyer have proved that under certain conditions a series summable by a Borel-type or absolute Borel-type method of summability is also summable by the Abel or absolute Abel method of summability to the same sum. In the present paper, the Abel method is replaced by the more general Abel-type method, giving similar results for ordinary, strong and absolute summability.

A Tauberian theorem for absolute summability

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 the relation between the Abel-type and Borel-type methods of summability

In recent papers, R. Jajte and B. L. R. Shawyer have proved that under certain conditions a series summable by a Borel-type or absolute Borel-type method of summability is also summable by the Abel or absolute Abel method of summability to the same sum. In the present paper, the Abel method is replaced by the more general Abel-type method, giving similar results for ordinary, strong and absolute summability.

On the Relation between the Abel and Borel-Type Methods of Summability

On the Relation between the Abel and Borel-Type Methods of Summability

On the relation between the Abel and Borel-type methods of summability.

In recent papers, R. Jajte and B. L. R. Shawyer have proved that under certain conditions a series summable by a Borel-type or absolute Borel-type method of summability is also summable by the Abel or absolute Abel method of summability to the same sum. In the present paper, the Abel method is replaced by the more general Abel-type method, giving similar results for ordinary, strong and absolute summability.

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

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

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.

A Note on Logarithmic Summability (L)

It is well known that two mutually related summability methods for a sequence sn(n = 0, 1, 2, …) are any Cesaro method (C, γ) of order γ > 0 and the Abel method (A). The notation used in this statement is that of Hardy ((1), pp. 96-7, 71) and the statement itself can be amplified as follows. Summability (C, γ), γ > 0, of sn implies (i) sn = o(nγ), (ii) summability (A) of sn. Also, as a conditional converse of this result, we have Offord's result ((3), first part of Theorem 2), that hypothesis (i), and hypothesis (ii) suitably strengthened, together imply summability (C, γ), γ > 0, of sn. It is the object of this note to bring to light a second pair of summability methods mutually related like the methods (C, γ) and (A), by following an argument which is essentially similar to Offord's but differs sufficiently from Offord's in details to justify a separate and self-contained treatment of the second pair of methods.

Evaluation of a Cholesterol "Kit"

Abstract A rapid "kit" method (Chole-Tech) for cholesterol determination in serum was evaluated by comparison with the Abell method in one clinical laboratory and with the Bloor method in another. Discrepancies between individual results of these standard technics and the Chole-Tech method were so great and so random that the results cannot be considered comparable. The Chole-Tech method is too unreliable for clinical use. All laboratory kits should be critically appraised before use. Accuracy and specificity must not be sacrificed for convenience.

The Taylor and other methods of summability

are regular if and only if 0 <q <1 as proved in [3] and [4]. The question of inclusion of T(r) and S(q) was discussed by Schoonmaker [4]. In ??2 and 3 it will be shown that both T(r) and S(q) include the binary method Y, [5], and that neither T(r) nor S(q) includes the Cesaro method C(1) or the Abel method A. ?4 consists of the determination of domains in which the geometric series Zn and the power series EanZn are summable by the iterated product method B* T(r), where B denotes the Borel exponential method. A summary of similar results for the product method B S(q) comprises ?5. The author is indebted to his referee for his several helpful comments and especially for his suggestions which led to simpler proofs of Theorems 3, 6 and 7.

On a scale of Abel-type summability methods

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.

An Analogue for Strong Summability of Abel's Summability Method

Given a series Σan, we define , by the relationwhere is the binomial coefficient . Let . If , the series Σan is said to be summable (C; k) to the sum s. If k &gt; 0, p ≥ 1 and if, as n → ∞,we say that the series Σan is summable [C; k, p] to the sum s, or that the series is strongly summable (C; k) with index p to the sum s. If denotes the difference , it is known that necessary and sufficient conditions for summability [C; k, p], k &gt; 0, p ≥ 1, to the sum s, are that Σan be summable (C; k) to the sum s and that

Some Inequalities Related to Abel's Method of Summation

Some Inequalities Related to Abel's Method of Summation

Notes on Linear Transformations. II. Analyticity of Semi-Groups

(1.1.4) T4[To x] = To+x a, f > 0. Here Wax is the Gauss-Weierstrass singular integral and Pax is the Poisson integral for the upper half-plane. Poisson's integral for the circle leads to the same functional equation provided we replace the customary parameter r by ea. The equation is also encountered in the applications of Abel's method of summation or its various generalizations (convergence factors e-aXn). As an example of a different type, leading to the same equation, we mention the theory of fractional integration.

The Application of Abel's Method of Summation to Dirichlet's Series

The Application of Abel's Method of Summation to Dirichlet's Series