# Summability, strong

of a complex sequence $\{ S _ {n} \}$ of numbers or functions (or of a series $\sum _ {k=} 1 ^ \infty a _ {k}$ with partial sums $S _ {n}$) to a number $S$

Summability by a method $A = | a _ {nk} |$( cf. Summation methods) such that for a certain $p > 0$:

1) the sequence

$$\sigma _ {n} = \sum _ { k= } 1 ^ \infty a _ {nk} | S _ {k} - S | ^ {p}$$

converges for every $n > 1$, and for almost all $x$ in the case of a sequence of functions;

2) $\lim\limits _ {n \rightarrow \infty } \sigma _ {n} = 0$. By retaining 2) and replacing 1) by:

1') for every monotone increasing sequence of indices $\{ v _ {k} \}$, the sequence

$$\sigma _ {n} = \sum _ { k= } 1 ^ \infty a _ {nv _ {k} } | S _ {v _ {k} } - S | ^ {p}$$

converges for every $n > 1$, and for almost all $x$ in the case of a sequence of functions, one arrives at the concept of very strong summability.

The concept of strong summability was introduced in connection with the $( C, 1)$- summability of Fourier series (cf. Summation of Fourier series). The importance of this concept is well illustrated in the example of strong $( C, 1)$- summability. Strong $( C, 1)$- summability signifies that the partial sums $S _ {\nu _ {1} } , S _ {\nu _ {2} } \dots$ that spoil the convergence of the sequence $\{ S _ {n} \}$ are sufficiently scarcely positioned, i.e. have zero density. Unlike strong summability, very strong summability means that the convergence of the sequence $\{ S _ {n} \}$ is spoiled by very thin sequences $\{ S _ {\nu _ {m} } \}$.

#### References

 [1] G.H. Hardy, J.E. Littlewood, "Sur la série de Fourier d'une fonction à carré sommable" C.R. Acad. Sci. Paris , 156 (1913) pp. 1307–1309 [2] G. Aleksich, "Convergence problems of orthogonal series" , Pergamon (1961) (Translated from Russian) [3] A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988) [4] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) [5] Gen-Ichirô Sunouchi, "Strong summability of Walsh–Fourier series" Tôhoku Math. J. , 16 (1964) pp. 228–237 [6] Gen-Ichirô Sunouchi, Acta Sci. Math. , 27 : 1–2 (1966) pp. 71–76 [7] V.A. Bolgov, E.V. Efimov, "On the rate of summability of orthogonal series" Math. USSR Izv. , 5 : 6 (1071) pp. 1399–1417 Izv. Akad. Nauk SSSR Ser. Mat. , 35 : 6 (1971) pp. 1389–1408 [8] Z. Zatewasser, Studia Math. , 6 (1936) pp. 82–88 [9] L. Leindler, "Ueber die sehr starke Riesz-Summierbarkeit der Orthogonalreihen und Konvergenz lückenhafter Orthogonalreihen" Acta Math. Acad. Sci. Hung. , 13 : 3–4 (1962) pp. 401–414
How to Cite This Entry:
Summability, strong. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Summability,_strong&oldid=48905
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article