Geodesic
Generalized Cesaro means of order −1
Glasgow mathematical journal, Tome 7 (1966) no. 3, pp. 119-124

Voir la notice de l'article provenant de la source Cambridge University Press

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 | < ∞. 
Maddox, I. J. Generalized Cesaro means of order −1. Glasgow mathematical journal, Tome 7 (1966) no. 3, pp. 119-124. doi: 10.1017/S2040618500035292
@article{10_1017_S2040618500035292,
     author = {Maddox, I. J.},
     title = {Generalized {Cesaro} means of order \ensuremath{-}1},
     journal = {Glasgow mathematical journal},
     pages = {119--124},
     year = {1966},
     volume = {7},
     number = {3},
     doi = {10.1017/S2040618500035292},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S2040618500035292/}
}
TY  - JOUR
AU  - Maddox, I. J.
TI  - Generalized Cesaro means of order −1
JO  - Glasgow mathematical journal
PY  - 1966
SP  - 119
EP  - 124
VL  - 7
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S2040618500035292/
DO  - 10.1017/S2040618500035292
ID  - 10_1017_S2040618500035292
ER  - 
%0 Journal Article
%A Maddox, I. J.
%T Generalized Cesaro means of order −1
%J Glasgow mathematical journal
%D 1966
%P 119-124
%V 7
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S2040618500035292/
%R 10.1017/S2040618500035292
%F 10_1017_S2040618500035292

[1] 1.Banach, S., Théorie des opérations linéaires (New York, 1955). Google Scholar

[2] 2.Chow, H. C., Note on convergence and summability factors, J. London Math. Soc. 29 (1954), 459–476. Google Scholar | DOI

[3] 3.Hardy, G. H. and Riesz, M., The general theory of Dirichlet's series (Cambridge Tract No. 18, 1915). Google Scholar

[4] 4.Knopp, K. and Lorentz, G. G., Beiträge zur absoluten Limitierung, Archiv. der Math. 2 (1949), 10–16. Google Scholar | DOI

[5] 5.Kuttner, B., On discontinuous Riesz means of type n, J. London Math. Soc. 37 (1962), 354–364. Google Scholar | DOI

[6] 6.Maddox, I. J., Matrix transformations of (C, -1) summable series, Proc. Koninkl. Nederl. Akad. van Wetenschappen A 68 (1965), 129–132. Google Scholar | DOI

[7] 7.Peyerimhoff, A., Summierbarkeitsfaktoren für absolut Cesàro-summierbare Reihen, Math. Z. 59 (1954), 417–424. Google Scholar | DOI

[8] 8.Young, W. H., On the convergence of the derived series of a Fourier series, Proc. London Math. Soc. (2) 17 (1918), 195–236. Google Scholar | DOI

Cité par Sources :