Summability, strong

of a complex sequence of numbers or functions (or of a series with partial sums ) to a number

Summability by a method (cf. Summation methods) such that for a certain :

1) the sequence

converges for every , and for almost all in the case of a sequence of functions;

2) . By retaining 2) and replacing 1) by:

1') for every monotone increasing sequence of indices , the sequence

converges for every , and for almost all 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 -summability of Fourier series (cf. Summation of Fourier series). The importance of this concept is well illustrated in the example of strong -summability. Strong -summability signifies that the partial sums that spoil the convergence of the sequence are sufficiently scarcely positioned, i.e. have zero density. Unlike strong summability, very strong summability means that the convergence of the sequence is spoiled by very thin sequences .

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