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A property of summation methods that describes the consistency of the results of applying these methods. Two methods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236602.png" /> are compatible if they cannot sum the same sequence or series to different limits, otherwise they are called incompatible summation methods. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236604.png" /> be summation methods of sequences say, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236606.png" /> be their summability fields. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236608.png" /> are compatible if
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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/c/c023/c023660/c0236609.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366012.png" /> are the numbers to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366013.png" /> is summed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366015.png" />, respectively. For example, all the [[Cesàro summation methods|Cesàro summation methods]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366016.png" /> are compatible for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366017.png" />, and so is every regular [[Voronoi summation method|Voronoi summation method]].
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A property of summation methods that describes the consistency of the results of applying these methods. Two methods  $  A $
 +
and $  B $
 +
are compatible if they cannot sum the same sequence or series to different limits, otherwise they are called incompatible summation methods. More precisely, let  $  A $
 +
and  $  B $
 +
be summation methods of sequences say, and let  $  A  ^ {*} $
 +
and  $  B  ^ {*} $
 +
be their summability fields. Then  $  A $
 +
and  $  B $
 +
are compatible if
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366018.png" /> is some set of sequences and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366019.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366020.png" />, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366022.png" /> are compatible on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366023.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366025.png" /> are said to be completely compatible (for real sequences) if (*) also holds in case one includes in their summability fields the sequences summable by these methods to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366027.png" />.
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$$ \tag{* }
 +
\overline{A}\; ( x)  = \
 +
\overline{B}\; ( x)
 +
$$
 +
 
 +
for any  $  x \in A  ^ {*} \cap B  ^ {*} $,
 +
where  $  \overline{A}\; ( x) $
 +
and  $  \overline{B}\; ( x) $
 +
are the numbers to which  $  x $
 +
is summed by  $  A $
 +
and  $  B $,
 +
respectively. For example, all the [[Cesàro summation methods|Cesàro summation methods]]  $  ( C , k ) $
 +
are compatible for  $  k > - 1 $,
 +
and so is every regular [[Voronoi summation method|Voronoi summation method]].
 +
 
 +
If  $  U $
 +
is some set of sequences and $  \overline{A}\; ( x) = \overline{B}\; ( x) $
 +
for every $  x \in A  ^ {*} \cap B  ^ {*} \cap U $,  
 +
then one says that $  A $
 +
and $  B $
 +
are compatible on $  U $.  
 +
$  A $
 +
and $  B $
 +
are said to be completely compatible (for real sequences) if (*) also holds in case one includes in their summability fields the sequences summable by these methods to $  + \infty $
 +
and $  - \infty $.
  
 
====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">  R.G. Cooke,  "Infinite matrices and sequence spaces" , Macmillan  (1950)</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">  R.G. Cooke,  "Infinite matrices and sequence spaces" , Macmillan  (1950)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 17:45, 4 June 2020


A property of summation methods that describes the consistency of the results of applying these methods. Two methods $ A $ and $ B $ are compatible if they cannot sum the same sequence or series to different limits, otherwise they are called incompatible summation methods. More precisely, let $ A $ and $ B $ be summation methods of sequences say, and let $ A ^ {*} $ and $ B ^ {*} $ be their summability fields. Then $ A $ and $ B $ are compatible if

$$ \tag{* } \overline{A}\; ( x) = \ \overline{B}\; ( x) $$

for any $ x \in A ^ {*} \cap B ^ {*} $, where $ \overline{A}\; ( x) $ and $ \overline{B}\; ( x) $ are the numbers to which $ x $ is summed by $ A $ and $ B $, respectively. For example, all the Cesàro summation methods $ ( C , k ) $ are compatible for $ k > - 1 $, and so is every regular Voronoi summation method.

If $ U $ is some set of sequences and $ \overline{A}\; ( x) = \overline{B}\; ( x) $ for every $ x \in A ^ {*} \cap B ^ {*} \cap U $, then one says that $ A $ and $ B $ are compatible on $ U $. $ A $ and $ B $ are said to be completely compatible (for real sequences) if (*) also holds in case one includes in their summability fields the sequences summable by these methods to $ + \infty $ and $ - \infty $.

References

[1] G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[2] R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)

Comments

The Voronoi summation methods are in the Western literature called Nörlund summation methods, despite the fact that G.F. Voronoi was the first to introduce these methods (1901). N.E. Nörlund later (1920), independently, rediscovered them.

References

[a1] K. Zeller, W. Beekmann, "Theorie der Limitierungsverfahren" , Springer (1970)
How to Cite This Entry:
Compatibility of summation methods. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Compatibility_of_summation_methods&oldid=11696
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article