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Numerical factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s0911001.png" /> (for the terms of a series) that transform a series
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<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/s/s091/s091100/s0911002.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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which is summable by a summation method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s0911003.png" /> (cf. [[Summation methods|Summation methods]]) into a series
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Numerical factors  $  \lambda _ {n} $(
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for the terms of a series) that transform a 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/s/s091/s091100/s0911004.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{1 }
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\sum _ { n= } 1 ^  \infty  u _ {n}  $$
  
which is summable by a method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s0911005.png" />. In this case, the summability multipliers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s0911006.png" /> are called summability multipliers of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s0911008.png" />. For example, the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s0911009.png" /> are summability multipliers of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s09110010.png" /> (see [[Cesàro summation methods|Cesàro summation methods]]) when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s09110011.png" /> (see [[#References|[1]]]).
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which is summable by a summation method $  A $(
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cf. [[Summation methods|Summation methods]]) into a series
  
The fundamental problem in the theory of summability multipliers is to find conditions under which numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s09110012.png" /> will be summability multipliers of one type or another. This question is formulated more exactly in the following way: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s09110013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s09110014.png" /> are two classes of series, then what conditions have to be imposed on the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s09110015.png" /> so that for every series (1) from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s09110016.png" />, the series (2) belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s09110017.png" />? The appearance of the theory of summability multipliers goes back to the Dedekind–Hadamard theorem: The series (2) converges for any convergent series (1) if and only if
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$$ \tag{2 }
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\sum _ { n= } 1 ^  \infty  \lambda _ {n} u _ {n}  $$
  
<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/s/s091/s091100/s09110018.png" /></td> </tr></table>
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which is summable by a method  $  B $.  
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In this case, the summability multipliers  $  \lambda _ {n} $
 +
are called summability multipliers of type  $  ( A, B) $.  
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For example, the numbers  $  \lambda _ {n} = 1/( n+ 1)  ^ {s} $
 +
are summability multipliers of type  $  (( C, k), ( C, k- s)) $(
 +
see [[Cesàro summation methods|Cesàro summation methods]]) when  $  0 < s < k+ 1 $(
 +
see [[#References|[1]]]).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091100/s09110019.png" />. There is a generalization of this theorem with summability by the Cesàro method.
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The fundamental problem in the theory of summability multipliers is to find conditions under which numbers  $  \lambda _ {n} $
 +
will be summability multipliers of one type or another. This question is formulated more exactly in the following way: If  $  X $
 +
and  $  Y $
 +
are two classes of series, then what conditions have to be imposed on the numbers  $  \lambda _ {n} $
 +
so that for every series (1) from  $  X $,
 +
the series (2) belongs to  $  Y $?
 +
The appearance of the theory of summability multipliers goes back to the Dedekind–Hadamard theorem: The series (2) converges for any convergent series (1) if and only if
 +
 
 +
$$
 +
\sum _ { n= } 0 ^  \infty  | \Delta \lambda _ {n} |  <  \infty ,
 +
$$
 +
 
 +
where  $  \Delta \lambda _ {n} = \lambda _ {n} - \lambda _ {n+} 1 $.  
 +
There is a generalization of this theorem with summability by the Cesàro method.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,  "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.F. Kangro,  "On summability factors"  ''Uchen. Zapiski Tartusk. Univ.'' , '''37'''  (1955)  pp. 191–232  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.F. Kangro,  "Theory of summability of sequences and series"  ''J. Soviet Math.'' , '''5''' :  1  (1970)  pp. 1–45  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''12'''  (1974)  pp. 5–70</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.A. Baron,  "Introduction to the theory of summability of series" , Tartu  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  C.N. Moore,  "Summable series and convergence factors" , Dover, reprint  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,  "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.F. Kangro,  "On summability factors"  ''Uchen. Zapiski Tartusk. Univ.'' , '''37'''  (1955)  pp. 191–232  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.F. Kangro,  "Theory of summability of sequences and series"  ''J. Soviet Math.'' , '''5''' :  1  (1970)  pp. 1–45  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''12'''  (1974)  pp. 5–70</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.A. Baron,  "Introduction to the theory of summability of series" , Tartu  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  C.N. Moore,  "Summable series and convergence factors" , Dover, reprint  (1966)</TD></TR></table>

Revision as of 08:24, 6 June 2020


Numerical factors $ \lambda _ {n} $( for the terms of a series) that transform a series

$$ \tag{1 } \sum _ { n= } 1 ^ \infty u _ {n} $$

which is summable by a summation method $ A $( cf. Summation methods) into a series

$$ \tag{2 } \sum _ { n= } 1 ^ \infty \lambda _ {n} u _ {n} $$

which is summable by a method $ B $. In this case, the summability multipliers $ \lambda _ {n} $ are called summability multipliers of type $ ( A, B) $. For example, the numbers $ \lambda _ {n} = 1/( n+ 1) ^ {s} $ are summability multipliers of type $ (( C, k), ( C, k- s)) $( see Cesàro summation methods) when $ 0 < s < k+ 1 $( see [1]).

The fundamental problem in the theory of summability multipliers is to find conditions under which numbers $ \lambda _ {n} $ will be summability multipliers of one type or another. This question is formulated more exactly in the following way: If $ X $ and $ Y $ are two classes of series, then what conditions have to be imposed on the numbers $ \lambda _ {n} $ so that for every series (1) from $ X $, the series (2) belongs to $ Y $? The appearance of the theory of summability multipliers goes back to the Dedekind–Hadamard theorem: The series (2) converges for any convergent series (1) if and only if

$$ \sum _ { n= } 0 ^ \infty | \Delta \lambda _ {n} | < \infty , $$

where $ \Delta \lambda _ {n} = \lambda _ {n} - \lambda _ {n+} 1 $. There is a generalization of this theorem with summability by the Cesàro method.

References

[1] G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[2] G.F. Kangro, "On summability factors" Uchen. Zapiski Tartusk. Univ. , 37 (1955) pp. 191–232 (In Russian)
[3] G.F. Kangro, "Theory of summability of sequences and series" J. Soviet Math. , 5 : 1 (1970) pp. 1–45 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 5–70
[4] S.A. Baron, "Introduction to the theory of summability of series" , Tartu (1966) (In Russian)
[5] C.N. Moore, "Summable series and convergence factors" , Dover, reprint (1966)
How to Cite This Entry:
Summability multipliers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Summability_multipliers&oldid=48906
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article