On Absolute Summability by Riesz and Generalized Cesàro means. II: Addendum

1971 ◽  
Vol 23 (5) ◽  
pp. 844-844
Author(s):  
Hans-Heinrich Körle
Keyword(s):  
Absolute Summability ◽  
Cesàro Means ◽  
Cesaro Means ◽  
Cesáro Means ◽  
Generalized Cesàro Means

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 Absolute Summability by Riesz and Generalized Cesàro Means. II

1970 ◽  
Vol 22 (2) ◽  
pp. 209-218 ◽  
Author(s):  
H.-H. Körle
Keyword(s):  
Absolute Summability ◽  
Wide Sense ◽  
Cesàro Means ◽  
Cesaro Means ◽  
Inclusion Relations ◽  
Cesáro Means ◽  
The Moment ◽  
Image Position ◽  
Generalized Cesàro Means

1. We will use the terminology of part I [9], including the general assumptions of [9, § 1]. In that paper we had proved that |R, λ, κ| = |C, λ, κ| in case that κ is an integer. Now, we turn to non-integral orders κ.As to ordinary summation, the following inclusion relations (in the customary sense; see [9, end of § 1]) for non-integral κ have been established so far. (Since we are comparing Riesz methods of the same type λ and order κ only, (R, λ, κ) is written (R), etc., for the moment.) (R) ⊆ (C) is a result by Borwein and Russell [2]. (C) ⊆ (R) was proved by Jurkat [3] in the case 0 < κ < 1, and, after Borwein [1], it holds in the case 1 < κ < 2 if(1)(2)(i.e. decreases in the wide sense).

On the Approximate Properties of Generalized Cesàro Means of Conjugate Trigonometric Fourier Series

2009 ◽  
Vol 16 (3) ◽  
pp. 413-425
Author(s):  
Teimuraz Akhobadze
Keyword(s):  
Fourier Series ◽  
Continuous Functions ◽  
Trigonometric Fourier Series ◽  
Cesàro Means ◽  
Space Of Continuous Functions ◽  
Cesaro Means ◽  
Cesáro Means ◽  
Generalized Cesàro Means

Abstract The behavior of generalized Cesàro (𝐶, α 𝑛)-means (α 𝑛 ∈ (–1, 0), 𝑛 = 1, 2, . . .) of conjugate trigonometric Fourier series of 𝐻𝑤 classes in the space of continuous functions is studied.

scholarly journals Generalized Cesaro means of order −1

1966 ◽  
Vol 7 (3) ◽  
pp. 119-124 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Bounded Variation ◽  
Equivalent Condition ◽  
Cesàro Means ◽  
Cesaro Means ◽  
Cesáro Means ◽  
Generalized Cesàro Means

A series ∑an is said to be summable (C, — 1) to s if it converges to s and nan = o(1) [8]. It is well known that this definition is equivalent to tn→s (n→∞), where tn = sn + nan, sn = ao + … + an. The series is summable | C, — 1 | to s if the sequence t = {tn} is of bounded variation (t ∈ B.V.), i.e. ∑ |; ▲tn |; = ∑ | tn - tn-1 | < ∞, and ∑ ▲tn = lim tn = s. An equivalent condition is ∑ | an |; < ∞, ∑an = s and ∑ | ▲(nan) | < ∞. For, suppose that ∑an = s | C, - 1 |. Since {sn} is the sequence of (C, 1)-means of {tn} and since | C, 0 | ⊂ | C, 1 |, we have ∑ | an | < ∞ and ∑an = s whence ∑ | ▲(nan) | < ∞. Conversely, ∑ | an | < ∞, ∑an = s and ∑ | ▲(nan) | < ∞ imply t ∈ B.V. and ∑▲n = s + lim nan. But lim nan = 0, since ∑ | an | < ∞.

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