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The set of points in the extended complex plane that for a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552401.png" /> is defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552402.png" /> be the smallest closed convex domain in the complex plane that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552403.png" />. If there is no half-plane containing these points, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552404.png" /> is the whole complex plane, including the point at infinity; if such half-planes exist, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552405.png" /> is their common part. The point at infinity belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552406.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552407.png" /> is unbounded, and does not if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552408.png" /> is bounded. The intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552409.png" /> is called the kernel of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524010.png" />.
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524011.png" /> is bounded, then its kernel coincides with the closed convex hull of the set of limit points; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524012.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524014.png" />, then the kernel is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524015.png" />. The kernel of a real sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524016.png" /> is the interval of the real line with end points:
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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/k/k055/k055240/k05524017.png" /></td> </tr></table>
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The set of points in the extended complex plane that for a sequence  $  \{ z _ {n} \} $
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is defined as follows. Let  $  R _ {n} $
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be the smallest closed convex domain in the complex plane that contains  $  z _ {n + 1 }  , z _ {n + 2 }  , .  .  . $.
 +
If there is no half-plane containing these points, then  $  R _ {n} $
 +
is the whole complex plane, including the point at infinity; if such half-planes exist, then  $  R _ {n} $
 +
is their common part. The point at infinity belongs to  $  R _ {n} $
 +
if  $  \{ z _ {n} \} $
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is unbounded, and does not if  $  \{ z _ {n} \} $
 +
is bounded. The intersection  $  K = \cap _ {n = 1 }  ^  \infty  R _ {n} $
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is called the kernel of the sequence  $  \{ z _ {n} \} $.
  
The kernel of a sequence cannot be empty, although it may consist only of the point at infinity, as, for example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524018.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524019.png" />. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524020.png" /> with kernel consisting of the point at infinity is sometimes called definitely divergent. For a real sequence this means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524021.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524022.png" />.
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If  $  \{ z _ {n} \} $
 +
is bounded, then its kernel coincides with the closed convex hull of the set of limit points; if  $  \{ z _ {n} \} $
 +
converges to  $  z _ {0} $,  
 +
$  z _ {0} \neq \infty $,  
 +
then the kernel is  $  z _ {0} $.  
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The kernel of a real sequence $  \{ z _ {n} \} $
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is the interval of the real line with end points:
  
Questions of kernel inclusion of summation methods are considered in the theory of summability. A summation method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524023.png" /> is kernel-stronger than a summation method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524024.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524025.png" /> of sequences if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524026.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524029.png" /> are, respectively, the kernels of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524031.png" />, that is, of sequences of averages of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524032.png" />.
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$$
 +
a  =  \overline{\lim\limits}\; _ {n \rightarrow \infty }  z _ {n} ,\ \
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b  =  \overline{\lim\limits}\; _ {n \rightarrow \infty }  z _ {n} .
 +
$$
 +
 
 +
The kernel of a sequence cannot be empty, although it may consist only of the point at infinity, as, for example, for  $  \{ z _ {n} \} $
 +
where  $  z _ {n} = n + in $.
 +
A sequence  $  \{ z _ {n} \} $
 +
with kernel consisting of the point at infinity is sometimes called definitely divergent. For a real sequence this means that  $  z _ {n} \rightarrow + \infty $
 +
or  $  z _ {n} \rightarrow - \infty $.
 +
 
 +
Questions of kernel inclusion of summation methods are considered in the theory of summability. A summation method $  A $
 +
is kernel-stronger than a summation method $  B $
 +
on a set $  U $
 +
of sequences if $  K _ {A} \subset  K _ {B} $
 +
for any $  \{ z _ {n} \} \subset  U $,  
 +
where $  K _ {A} $
 +
and $  K _ {B} $
 +
are, respectively, the kernels of $  A $
 +
and $  B $,  
 +
that is, of sequences of averages of $  \{ z _ {n} \} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  K. Knopp,  "Zur Theorie des Limitierungsverfahren I"  ''Math. Z.'' , '''31'''  (1930)  pp. 97–127</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  K. Knopp,  "Zur Theorie des Limitierungsverfahren II"  ''Math. Z.'' , '''31'''  (1930)  pp. 276–305</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">  G.H. Hardy,  "Divergent series" , Clarendon Press  (1949)</TD></TR></table>
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  K. Knopp,  "Zur Theorie des Limitierungsverfahren I"  ''Math. Z.'' , '''31'''  (1930)  pp. 97–127</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  K. Knopp,  "Zur Theorie des Limitierungsverfahren II"  ''Math. Z.'' , '''31'''  (1930)  pp. 276–305</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">  G.H. Hardy,  "Divergent series" , Clarendon Press  (1949)</TD></TR></table>

Latest revision as of 22:14, 5 June 2020


The set of points in the extended complex plane that for a sequence $ \{ z _ {n} \} $ is defined as follows. Let $ R _ {n} $ be the smallest closed convex domain in the complex plane that contains $ z _ {n + 1 } , z _ {n + 2 } , . . . $. If there is no half-plane containing these points, then $ R _ {n} $ is the whole complex plane, including the point at infinity; if such half-planes exist, then $ R _ {n} $ is their common part. The point at infinity belongs to $ R _ {n} $ if $ \{ z _ {n} \} $ is unbounded, and does not if $ \{ z _ {n} \} $ is bounded. The intersection $ K = \cap _ {n = 1 } ^ \infty R _ {n} $ is called the kernel of the sequence $ \{ z _ {n} \} $.

If $ \{ z _ {n} \} $ is bounded, then its kernel coincides with the closed convex hull of the set of limit points; if $ \{ z _ {n} \} $ converges to $ z _ {0} $, $ z _ {0} \neq \infty $, then the kernel is $ z _ {0} $. The kernel of a real sequence $ \{ z _ {n} \} $ is the interval of the real line with end points:

$$ a = \overline{\lim\limits}\; _ {n \rightarrow \infty } z _ {n} ,\ \ b = \overline{\lim\limits}\; _ {n \rightarrow \infty } z _ {n} . $$

The kernel of a sequence cannot be empty, although it may consist only of the point at infinity, as, for example, for $ \{ z _ {n} \} $ where $ z _ {n} = n + in $. A sequence $ \{ z _ {n} \} $ with kernel consisting of the point at infinity is sometimes called definitely divergent. For a real sequence this means that $ z _ {n} \rightarrow + \infty $ or $ z _ {n} \rightarrow - \infty $.

Questions of kernel inclusion of summation methods are considered in the theory of summability. A summation method $ A $ is kernel-stronger than a summation method $ B $ on a set $ U $ of sequences if $ K _ {A} \subset K _ {B} $ for any $ \{ z _ {n} \} \subset U $, where $ K _ {A} $ and $ K _ {B} $ are, respectively, the kernels of $ A $ and $ B $, that is, of sequences of averages of $ \{ z _ {n} \} $.

References

[1a] K. Knopp, "Zur Theorie des Limitierungsverfahren I" Math. Z. , 31 (1930) pp. 97–127
[1b] K. Knopp, "Zur Theorie des Limitierungsverfahren II" Math. Z. , 31 (1930) pp. 276–305
[2] R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)
[3] G.H. Hardy, "Divergent series" , Clarendon Press (1949)
How to Cite This Entry:
Kernel of a complex sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kernel_of_a_complex_sequence&oldid=13731
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article