Extension of a theorem of Sunouchi

1969 ◽  
Vol 65 (2) ◽  
pp. 489-494 ◽  
Author(s):  
V. P. Srivastava
Keyword(s):  
Infinite Series ◽  
Cesàro Means ◽  
Cesaro Means ◽  
Partial Sum ◽  
Cesáro Means

1·1. Definitions and notations. Let ∑an be a given infinite series, and let sn be its nth partial sum. We denote by and the nth Cesàro means of order α(α > − 1) of the sequences {sn} and {n. an} respectively.

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 → ∞.

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

scholarly journals Lebesgue points of $$\ell _1$$-Cesàro summability of d-dimensional Fourier series

2021 ◽  
Vol 6 (3) ◽  
Author(s):  
Ferenc Weisz
Keyword(s):  
Fourier Series ◽  
Lebesgue Point ◽  
Cesàro Means ◽  
Cesàro Summability ◽  
Cesaro Means ◽  
Cesaro Summability ◽  
Lebesgue Points ◽  
Cesáro Means

AbstractWe generalize the classical Lebesgue’s theorem and prove that the $$\ell _1$$ ℓ 1 -Cesàro means of the Fourier series of the multi-dimensional function $$f\in L_1({{\mathbb {T}}}^d)$$ f ∈ L 1 ( T d ) converge to f at each strong $$\omega $$ ω -Lebesgue point.

Growth conditions on Cesàro means of higher order

2013 ◽  
Vol 79 (3-4) ◽  
pp. 545-581
Author(s):  
Laurian Suciu ◽  
Jaroslav Zemánek
Keyword(s):  
Growth Conditions ◽  
Higher Order ◽  
Cesàro Means ◽  
Cesaro Means ◽  
Cesáro Means

Uniform summability of power series

1972 ◽  
Vol 71 (2) ◽  
pp. 335-341 ◽  
Author(s):  
J. C. Kurtz ◽  
W. T. Sledd
Keyword(s):  
Banach Space ◽  
Power Series ◽  
Normal Matrix ◽  
Cesàro Means ◽  
Summability Factors ◽  
Cesaro Means ◽  
Space Topology ◽  
Nörlund Means ◽  
Summability Field ◽  
Cesáro Means

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.

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