View source for Unconditional summability

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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|Summation methods]]). The study of unconditional summability originated with W. Orlicz [[#References|[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|Regular summation methods]]) implies [[Unconditional convergence|unconditional convergence]]. It was subsequently shown that this condition may be replaced by a weaker one: $\varliminf_{n\to\infty}a_n=0$ [[#References|[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|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 $\sum_{n=1}^\infty1$ [[#References|[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 $\sum_{n=1}^\infty1$ [[#References|[2]]].

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



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Zeller,   W. Beekmann,   "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>