# 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 $\{ S _ {\nu _ {m} } \}$.

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