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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553101.png" /> (depending on a parameter) the values of which are the averages of the given method of summation applied to the 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/k/k055/k055310/k0553102.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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The kernel of a summation method gives an integral representation of the averages of the method in the summation of [[Fourier series|Fourier series]]. If the summation method is defined by a transformation of sequences into sequences using a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553103.png" />, then the kernel of this method is the function
+
A function  $  K _ {n} ( t) $(
 +
depending on a parameter) the values of which are the averages of the given method of summation applied to 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/k/k055/k055310/k0553104.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
{
 +
\frac{1}{2}
 +
} + \sum _ {\nu = 1 } ^  \infty  \cos  \nu t.
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553105.png" /> are the partial sums of the series (1):
+
The kernel of a summation method gives an integral representation of the averages of the method in the summation of [[Fourier series|Fourier series]]. If the summation method is defined by a transformation of sequences into sequences using a matrix  $  \| a _ {nk} \| _ {n,k= 0 }  ^  \infty  $,
 +
then the kernel of this method is the function
  
<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/k055310/k0553106.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$
 +
K _ {n} ( t)  = \sum _ {k = 0 } ^  \infty  a _ {nk} D _ {k} ( t),
 +
$$
  
In this case the averages of the Fourier series for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553107.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553108.png" /> can be expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k0553109.png" /> and the kernel by the formula
+
where  $  D _ {k} ( t) $
 +
are the partial sums of the series (1):
  
<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/k055310/k05531010.png" /></td> </tr></table>
+
$$ \tag{2 }
 +
D _ {k} ( t)  = \
 +
{
 +
\frac{1}{2}
 +
} + \sum _ {\nu = 1 } ^ { k }  \cos  \nu t  = \
 +
 
 +
\frac{\sin \{ ( k + {1 / 2 } ) t \} }{2  \sin  ( t / 2) }
 +
.
 +
$$
 +
 
 +
In this case the averages of the Fourier series for a  $  2 \pi $-
 +
periodic function  $  f $
 +
can be expressed in terms of  $  f $
 +
and the kernel by the formula
 +
 
 +
$$
 +
\sigma _ {n} ( f, x)  = \
 +
{
 +
\frac{1} \pi
 +
}
 +
\int\limits _ {- \pi } ^  \pi  f ( t) K _ {n} ( t - x)  dt.
 +
$$
  
 
In particular, the kernel of the method of arithmetical averages (cf. [[Arithmetical averages, summation method of|Arithmetical averages, summation method of]]) has the form
 
In particular, the kernel of the method of arithmetical averages (cf. [[Arithmetical averages, summation method of|Arithmetical averages, summation method of]]) has 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/k/k055/k055310/k05531011.png" /></td> </tr></table>
+
$$
 +
K _ {n} ( t)  = \
 +
 
 +
\frac{2}{n + 1 }
 +
 
 +
\left [
 +
 
 +
\frac{\sin \{ ( n + 1) t /2 \} }{2  \sin ( t/2) }
 +
 
 +
\right ]  ^ {2} ,
 +
$$
  
 
and is called the Fejér kernel. The kernel of the [[Abel summation method|Abel summation method]] is given by
 
and is called the Fejér kernel. The kernel of the [[Abel summation method|Abel summation method]] is given by
  
<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/k055310/k05531012.png" /></td> </tr></table>
+
$$
 +
K ( r, t)  = \
 +
{
 +
\frac{1}{2}
 +
}
 +
 
 +
\frac{1 - r  ^ {2} }{1 - 2r  \cos  t + r  ^ {2} }
 +
,\ \
 +
0 \leq  r < 1,
 +
$$
  
and is called the Poisson kernel. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k05531013.png" /> in (2) is called the Dirichlet kernel.
+
and is called the Poisson kernel. The function $  D _ {k} ( t) $
 +
in (2) is called the Dirichlet kernel.
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055310/k05531014.png" /> whose values are the averages of a summation method applied to the series
+
The function $  \overline{K}\; _ {n} ( t) $
 +
whose values are the averages of a summation method applied to 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/k/k055/k055310/k05531015.png" /></td> </tr></table>
+
$$
 +
\sum _ {\nu = 1 } ^  \infty  \sin  \nu t
 +
$$
  
 
is called the conjugate kernel of the summation method.
 
is called the conjugate kernel of the summation method.

Latest revision as of 22:14, 5 June 2020


A function $ K _ {n} ( t) $( depending on a parameter) the values of which are the averages of the given method of summation applied to the series

$$ \tag{1 } { \frac{1}{2} } + \sum _ {\nu = 1 } ^ \infty \cos \nu t. $$

The kernel of a summation method gives an integral representation of the averages of the method in the summation of Fourier series. If the summation method is defined by a transformation of sequences into sequences using a matrix $ \| a _ {nk} \| _ {n,k= 0 } ^ \infty $, then the kernel of this method is the function

$$ K _ {n} ( t) = \sum _ {k = 0 } ^ \infty a _ {nk} D _ {k} ( t), $$

where $ D _ {k} ( t) $ are the partial sums of the series (1):

$$ \tag{2 } D _ {k} ( t) = \ { \frac{1}{2} } + \sum _ {\nu = 1 } ^ { k } \cos \nu t = \ \frac{\sin \{ ( k + {1 / 2 } ) t \} }{2 \sin ( t / 2) } . $$

In this case the averages of the Fourier series for a $ 2 \pi $- periodic function $ f $ can be expressed in terms of $ f $ and the kernel by the formula

$$ \sigma _ {n} ( f, x) = \ { \frac{1} \pi } \int\limits _ {- \pi } ^ \pi f ( t) K _ {n} ( t - x) dt. $$

In particular, the kernel of the method of arithmetical averages (cf. Arithmetical averages, summation method of) has the form

$$ K _ {n} ( t) = \ \frac{2}{n + 1 } \left [ \frac{\sin \{ ( n + 1) t /2 \} }{2 \sin ( t/2) } \right ] ^ {2} , $$

and is called the Fejér kernel. The kernel of the Abel summation method is given by

$$ K ( r, t) = \ { \frac{1}{2} } \frac{1 - r ^ {2} }{1 - 2r \cos t + r ^ {2} } ,\ \ 0 \leq r < 1, $$

and is called the Poisson kernel. The function $ D _ {k} ( t) $ in (2) is called the Dirichlet kernel.

The function $ \overline{K}\; _ {n} ( t) $ whose values are the averages of a summation method applied to the series

$$ \sum _ {\nu = 1 } ^ \infty \sin \nu t $$

is called the conjugate kernel of the summation method.

References

[1] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[2] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
How to Cite This Entry:
Kernel of a summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kernel_of_a_summation_method&oldid=47490
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article