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