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A summation method (cf. [[Summation methods|Summation methods]]) for series and sequences, defined by means of sequences of functions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840302.png" /> be a sequence of functions defined on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840303.png" /> of variation of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840304.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840305.png" /> be an [[Accumulation point|accumulation point]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840306.png" /> (finite or infinite). The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840307.png" /> are used to convert a given sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840308.png" /> into a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s0840309.png" />:
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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/s084/s084030/s08403010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|done}}
  
If the series in (1) is convergent for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403011.png" /> sufficiently close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403012.png" />, and if
+
A summation method (cf. [[Summation methods|Summation methods]]) for series and sequences, defined by means of sequences of functions. Let  $  \{ a _ {k} ( \omega ) \} $,
 +
$  k = 0 , 1 \dots $
 +
be a sequence of functions defined on some set  $  E $
 +
of variation of the parameter  $  \omega $,
 +
and let  $  \omega _ {0} $
 +
be an [[Accumulation point|accumulation point]] of  $  E $(
 +
finite or infinite). The functions  $  a _ {k} ( \omega ) $
 +
are used to convert a given sequence  $  \{ s _ {n} \} $
 +
into a function  $  \sigma ( \omega ) $:
  
<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/s084/s084030/s08403013.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
\sigma ( \omega )  = \sum _ { k= } 0 ^  \infty 
 +
a _ {k} ( \omega ) s _ {k} .
 +
$$
  
one says that the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403014.png" /> is summable to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403015.png" /> by the semi-continuous summation method defined by the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403017.png" /> is the sequence of partial sums of the series
+
If the series in (1) is convergent for all  $  \omega $
 +
sufficiently close to $  \omega _ {0} $,
 +
and if
  
<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/s084/s084030/s08403018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$
 +
\lim\limits _ {\omega \rightarrow \omega _ {0} }  \sigma ( \omega )  = s ,
 +
$$
  
one says that the series (2) is summable by the semi-continuous method to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403019.png" />. A semi-continuous summation method with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403020.png" /> is an analogue of the [[Matrix summation method|matrix summation method]] defined by the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403021.png" />, in which the integer-valued parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403022.png" /> is replaced by the continuous parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403023.png" />. The sequence of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403024.png" /> is therefore known as a semi-continuous matrix.
+
one says that the sequence  $  \{ s _ {n} \} $
 +
is summable to  $  s $
 +
by the semi-continuous summation method defined by the sequence  $  \{ a _ {k} ( \omega ) \} $.  
 +
If  $  \{ s _ {n} \} $
 +
is the sequence of partial sums of the series
  
A semi-continuous summation method can be defined by direct transformation of a series into a function, using a given sequence of functions, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403025.png" />:
+
$$ \tag{2 }
 +
\sum _ { k= } 0 ^  \infty  u _ {k} ,
 +
$$
  
<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/s084/s084030/s08403026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
one says that the series (2) is summable by the semi-continuous method to  $  s $.  
 +
A semi-continuous summation method with  $  \omega _ {0} = \infty $
 +
is an analogue of the [[Matrix summation method|matrix summation method]] defined by the matrix  $  \| a _ {nk} \| $,
 +
in which the integer-valued parameter  $  n $
 +
is replaced by the continuous parameter  $  \omega $.
 +
The sequence of functions  $  a _ {k} ( \omega ) $
 +
is therefore known as a semi-continuous matrix.
  
In this case the series (2) is said to be summable to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403027.png" /> if
+
A semi-continuous summation method can be defined by direct transformation of a series into a function, using a given sequence of functions, say  $  \{ g _ {k} ( \omega ) \} $:
  
<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/s084/s084030/s08403028.png" /></td> </tr></table>
+
$$ \tag{3 }
 +
\gamma ( \omega )  = \sum _ { k= } 0 ^  \infty  g _ {k} ( \omega ) u _ {k} .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403029.png" /> is an accumulation point of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403030.png" /> of variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403031.png" />, and the series (3) is assumed to be convergent for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403032.png" /> sufficiently close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403033.png" />.
+
In this case the series (2) is said to be summable to  $  s $
 +
if
 +
 
 +
$$
 +
\lim\limits _ {\omega \rightarrow \omega _ {0} }  \gamma ( \omega )  = s ,
 +
$$
 +
 
 +
where  $  \omega _ {0} $
 +
is an accumulation point of the set $  E $
 +
of variation of $  \omega $,  
 +
and the series (3) is assumed to be convergent for all $  \omega $
 +
sufficiently close to $  \omega _ {0} $.
  
 
In some cases a semi-continuous summation method is more convenient than a summation method based on ordinary matrices, since it enables one to utilize tools of function theory. Examples of semi-continuous summation methods are: the [[Abel summation method|Abel summation method]], the [[Borel summation method|Borel summation method]], the [[Lindelöf summation method|Lindelöf summation method]], and the [[Mittag-Leffler summation method|Mittag-Leffler summation method]]. The class of semi-continuous methods also includes methods with semi-continuous matrices of the form
 
In some cases a semi-continuous summation method is more convenient than a summation method based on ordinary matrices, since it enables one to utilize tools of function theory. Examples of semi-continuous summation methods are: the [[Abel summation method|Abel summation method]], the [[Borel summation method|Borel summation method]], the [[Lindelöf summation method|Lindelöf summation method]], and the [[Mittag-Leffler summation method|Mittag-Leffler summation method]]. The class of semi-continuous methods also includes methods with semi-continuous matrices of the form
  
<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/s084/s084030/s08403034.png" /></td> </tr></table>
+
$$
 +
a _ {k} ( \omega )  =
 +
\frac{p _ {k} \omega  ^ {k} }{\sum _ { l= } 0 ^  \infty 
 +
p _ {l} \omega  ^ {l} }
 +
,
 +
$$
  
 
where the denominator is an entire function that does not reduce to a polynomial.
 
where the denominator is an entire function that does not reduce to a polynomial.
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Conditions for the regularity of semi-continuous summation methods are analogous to regularity conditions for matrix summation methods. For example, the conditions
 
Conditions for the regularity of semi-continuous summation methods are analogous to regularity conditions for matrix summation methods. For example, the conditions
  
<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/s084/s084030/s08403035.png" /></td> </tr></table>
+
$$
 +
\sum _ { k= } 0 ^  \infty  | a _ {k} ( \omega ) |  \leq  M
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403036.png" /> sufficiently close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403037.png" />,
+
for all $  \omega $
 +
sufficiently close to $  \omega _ {0} $,
  
<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/s084/s084030/s08403038.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\omega \rightarrow \omega _ {0} }  a _ {k} ( \omega )  = 0 ,\ \
 +
k = 0 , 1 \dots
 +
$$
  
<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/s084/s084030/s08403039.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\omega \rightarrow \omega _ {0} }  \sum _ { k= } 0 ^  \infty  a _ {k} ( \omega )  = 1
 +
$$
  
are necessary and sufficient for the semi-continuous summation method defined by the transformation (1) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084030/s08403040.png" /> into a function to be regular (see [[Regularity criteria|Regularity criteria]]).
+
are necessary and sufficient for the semi-continuous summation method defined by the transformation (1) of $  \{ s _ {k} \} $
 +
into a function to be regular (see [[Regularity criteria|Regularity criteria]]).
  
 
====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><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Beekmann,  K. Zeller,  "Theorie der Limitierungsverfahren" , Springer  (1970)</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><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Beekmann,  K. Zeller,  "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>

Latest revision as of 08:13, 6 June 2020


A summation method (cf. Summation methods) for series and sequences, defined by means of sequences of functions. Let $ \{ a _ {k} ( \omega ) \} $, $ k = 0 , 1 \dots $ be a sequence of functions defined on some set $ E $ of variation of the parameter $ \omega $, and let $ \omega _ {0} $ be an accumulation point of $ E $( finite or infinite). The functions $ a _ {k} ( \omega ) $ are used to convert a given sequence $ \{ s _ {n} \} $ into a function $ \sigma ( \omega ) $:

$$ \tag{1 } \sigma ( \omega ) = \sum _ { k= } 0 ^ \infty a _ {k} ( \omega ) s _ {k} . $$

If the series in (1) is convergent for all $ \omega $ sufficiently close to $ \omega _ {0} $, and if

$$ \lim\limits _ {\omega \rightarrow \omega _ {0} } \sigma ( \omega ) = s , $$

one says that the sequence $ \{ s _ {n} \} $ is summable to $ s $ by the semi-continuous summation method defined by the sequence $ \{ a _ {k} ( \omega ) \} $. If $ \{ s _ {n} \} $ is the sequence of partial sums of the series

$$ \tag{2 } \sum _ { k= } 0 ^ \infty u _ {k} , $$

one says that the series (2) is summable by the semi-continuous method to $ s $. A semi-continuous summation method with $ \omega _ {0} = \infty $ is an analogue of the matrix summation method defined by the matrix $ \| a _ {nk} \| $, in which the integer-valued parameter $ n $ is replaced by the continuous parameter $ \omega $. The sequence of functions $ a _ {k} ( \omega ) $ is therefore known as a semi-continuous matrix.

A semi-continuous summation method can be defined by direct transformation of a series into a function, using a given sequence of functions, say $ \{ g _ {k} ( \omega ) \} $:

$$ \tag{3 } \gamma ( \omega ) = \sum _ { k= } 0 ^ \infty g _ {k} ( \omega ) u _ {k} . $$

In this case the series (2) is said to be summable to $ s $ if

$$ \lim\limits _ {\omega \rightarrow \omega _ {0} } \gamma ( \omega ) = s , $$

where $ \omega _ {0} $ is an accumulation point of the set $ E $ of variation of $ \omega $, and the series (3) is assumed to be convergent for all $ \omega $ sufficiently close to $ \omega _ {0} $.

In some cases a semi-continuous summation method is more convenient than a summation method based on ordinary matrices, since it enables one to utilize tools of function theory. Examples of semi-continuous summation methods are: the Abel summation method, the Borel summation method, the Lindelöf summation method, and the Mittag-Leffler summation method. The class of semi-continuous methods also includes methods with semi-continuous matrices of the form

$$ a _ {k} ( \omega ) = \frac{p _ {k} \omega ^ {k} }{\sum _ { l= } 0 ^ \infty p _ {l} \omega ^ {l} } , $$

where the denominator is an entire function that does not reduce to a polynomial.

Conditions for the regularity of semi-continuous summation methods are analogous to regularity conditions for matrix summation methods. For example, the conditions

$$ \sum _ { k= } 0 ^ \infty | a _ {k} ( \omega ) | \leq M $$

for all $ \omega $ sufficiently close to $ \omega _ {0} $,

$$ \lim\limits _ {\omega \rightarrow \omega _ {0} } a _ {k} ( \omega ) = 0 ,\ \ k = 0 , 1 \dots $$

$$ \lim\limits _ {\omega \rightarrow \omega _ {0} } \sum _ { k= } 0 ^ \infty a _ {k} ( \omega ) = 1 $$

are necessary and sufficient for the semi-continuous summation method defined by the transformation (1) of $ \{ s _ {k} \} $ into a function to be regular (see Regularity criteria).

References

[1] G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[2] R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)
[3] W. Beekmann, K. Zeller, "Theorie der Limitierungsverfahren" , Springer (1970)
How to Cite This Entry:
Semi-continuous summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-continuous_summation_method&oldid=48657
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article