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Difference between revisions of "Unconditional summability"

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Summability of a series for all possible rearrangements of its terms. The series
 
Summability of a series for all possible rearrangements of its terms. The series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u0951201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\sum_{n=1}^\infty a_n\tag{*}$$
  
is called unconditionally summable by some summation method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u0951202.png" /> (unconditionally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u0951203.png" />-summable) if it is summable by this method to a sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u0951204.png" /> whatever the ordering of its terms, where the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u0951205.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u0951206.png" />, 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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u0951207.png" /> [[#References|[2]]]. Unconditional summability by a matrix method does not imply unconditional convergence; in fact, take the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u0951208.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u0951209.png" /> is a regular [[Matrix summation method|matrix summation method]] and if the series (*) is unconditionally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512010.png" />-summable, then all its terms have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512012.png" /> is a constant and the series with terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512013.png" /> is absolutely convergent: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512014.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512015.png" /> if the method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512016.png" /> does not sum the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512017.png" /> [[#References|[3]]].
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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 \ref{*} 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512018.png" /> of measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512019.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512020.png" /> is unconditionally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512021.png" />-summable almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512022.png" />, then the terms of this series have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512024.png" /> is a finite measurable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512025.png" /> and the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512026.png" /> is unconditionally almost-everywhere convergent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512027.png" />; also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512028.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512029.png" /> does not sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095120/u09512030.png" /> [[#References|[2]]].
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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====
 
====References====

Revision as of 16:26, 3 June 2016

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

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

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 \ref{*} 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


Comments

References

[a1] K. Zeller, W. Beekmann, "Theorie der Limitierungsverfahren" , Springer (1970)
How to Cite This Entry:
Unconditional summability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unconditional_summability&oldid=12649
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article