Unconditional summability

Summability of a series for all possible rearrangements of its terms. The series

$$\sum_{n=1}^\infty a_n\label{*}\tag{*}$$

is called unconditionally summable by some summation method $A$ (unconditionally $A$-summable) if it is summable by this method to a sum $s$ whatever the ordering of its terms, where the value of $s$ may depend on the particular rearrangement (cf. Summation methods). The study of unconditional summability originated with W. Orlicz [1]; he showed, in particular, that if $\lim_{n\to\infty}a_n=0$, then absolute summability of the series by a linear regular method (cf. Regular summation methods) implies unconditional convergence. It was subsequently shown that this condition may be replaced by a weaker one: $\varliminf_{n\to\infty}a_n=0$ [2]. Unconditional summability by a matrix method does not imply unconditional convergence; in fact, take the series $\sum_{n=1}^\infty1$. If $A$ is a regular matrix summation method and if the series $\eqref{*}$ is unconditionally $A$-summable, then all its terms have the form $a_n=c+\eta_n$, where $c$ is a constant and the series with terms $\eta_n$ is absolutely convergent: $\sum_{n=1}^\infty|\eta_n|<\infty$; moreover, $c=0$ if the method $A$ does not sum the series $\sum_{n=1}^\infty1$ [3].

In the case of series of functions one distinguishes between summability in measure, everywhere summability, almost-everywhere summability, etc. For unconditional summability of a series of functions, the following statement is valid almost-everywhere: If the series $\sum_{n=1}^\infty f_n(x)$ of measurable functions $f_n$ on a set $E$ is unconditionally $A$-summable almost-everywhere on $E$, then the terms of this series have the form $f_n(x)=f(x)+\eta_n(x)$, where $f$ is a finite measurable function on $E$ and the series $\sum_{n=1}^\infty\eta_n(x)$ is unconditionally almost-everywhere convergent on $E$; also, $f=0$ if $A$ does not sum $\sum_{n=1}^\infty1$ [2].

References

[1]	W. Orlicz, Bull. Acad. Polon. Sci. Sér. Sci. Math., Astr. Phys. : 3A (1927) pp. 117–125
[2]	P.L. Ul'yanov, "Unconditional summability" Izv. Akad. Nauk SSSR Ser. Mat. , 23 (1959) pp. 781–808 (In Russian)
[3]	V.F. Gaposhkin, A.M. Olevskii, Nauchn. Dokl. Vyssh. Shkoly Fiz.-Mat. Nauk. , 6 (1958) pp. 81–86

References

[a1]	K. Zeller, W. Beekmann, "Theorie der Limitierungsverfahren" , Springer (1970)