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One of the arithmetic means of the partial sums of a Fourier series in the trigonometric system
 
One of the arithmetic means of the partial sums of a Fourier series in the trigonometric system
  
<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/f/f038/f038360/f0383601.png" /></td> </tr></table>
+
$$
 +
\sigma _ {n} ( f, x)  = \
 +
{
 +
\frac{1}{n + 1 }
 +
}
 +
\sum _ {k = 0 } ^ { n }
 +
s _ {k} ( f, x) =
 +
$$
  
<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/f/f038/f038360/f0383602.png" /></td> </tr></table>
+
$$
 +
= \
 +
{
 +
\frac{a _ {0} }{2}
 +
} + \sum _ {k = 1 } ^ { n }  \left ( 1 - {
 +
\frac{k}{n + 1 }
 +
} \right ) ( a _ {k}  \cos  kx + b _ {k}  \sin  kx),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038360/f0383603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038360/f0383604.png" /> are the Fourier coefficients of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038360/f0383605.png" />.
+
where $  a _ {k} $
 +
and $  b _ {k} $
 +
are the Fourier coefficients of the function f $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038360/f0383606.png" /> is continuous, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038360/f0383607.png" /> converges uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038360/f0383608.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038360/f0383609.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038360/f03836010.png" /> in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038360/f03836011.png" />.
+
If f $
 +
is continuous, then $  \sigma _ {n} ( f, x) $
 +
converges uniformly to f ( x) $;  
 +
$  \sigma _ {n} ( f, x) $
 +
converges to f ( x) $
 +
in the metric of $  L $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038360/f03836012.png" /> belongs to the class of functions that satisfy a Lipschitz condition of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038360/f03836013.png" />, then
+
If f $
 +
belongs to the class of functions that satisfy a Lipschitz condition of order $  \alpha < 1 $,  
 +
then
  
<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/f/f038/f038360/f03836014.png" /></td> </tr></table>
+
$$
 +
\| f ( x) - \sigma _ {n} ( f, x) \| _ {c}  = \
 +
O \left ( {
 +
\frac{1}{n  ^  \alpha  }
 +
} \right ) ,
 +
$$
  
that is, in this case the Fejér sum approximates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038360/f03836015.png" /> at the rate of the best approximating functions of the indicated class. But Fejér sums cannot provide a high rate of approximation: The estimate
+
that is, in this case the Fejér sum approximates f $
 +
at the rate of the best approximating functions of the indicated class. But Fejér sums cannot provide a high rate of approximation: The estimate
  
<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/f/f038/f038360/f03836016.png" /></td> </tr></table>
+
$$
 +
\| f ( x) - \sigma _ {n} ( f, x) \| _ {c}  = \
 +
o \left ( {
 +
\frac{1}{n}
 +
} \right )
 +
$$
  
 
is valid only for constant functions.
 
is valid only for constant functions.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Fejér,  "Untersuchungen über Fouriersche Reihen"  ''Math. Ann.'' , '''58'''  (1903)  pp. 51–69</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,  "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.P. Natanson,  "Constructive function theory" , '''1–3''' , F. Ungar  (1964–1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.M. Tikhomirov,  "Some problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Fejér,  "Untersuchungen über Fouriersche Reihen"  ''Math. Ann.'' , '''58'''  (1903)  pp. 51–69</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,  "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.P. Natanson,  "Constructive function theory" , '''1–3''' , F. Ungar  (1964–1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.M. Tikhomirov,  "Some problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
See also [[Fejér summation method|Fejér summation method]].
 
See also [[Fejér summation method|Fejér summation method]].

Latest revision as of 19:38, 5 June 2020


One of the arithmetic means of the partial sums of a Fourier series in the trigonometric system

$$ \sigma _ {n} ( f, x) = \ { \frac{1}{n + 1 } } \sum _ {k = 0 } ^ { n } s _ {k} ( f, x) = $$

$$ = \ { \frac{a _ {0} }{2} } + \sum _ {k = 1 } ^ { n } \left ( 1 - { \frac{k}{n + 1 } } \right ) ( a _ {k} \cos kx + b _ {k} \sin kx), $$

where $ a _ {k} $ and $ b _ {k} $ are the Fourier coefficients of the function $ f $.

If $ f $ is continuous, then $ \sigma _ {n} ( f, x) $ converges uniformly to $ f ( x) $; $ \sigma _ {n} ( f, x) $ converges to $ f ( x) $ in the metric of $ L $.

If $ f $ belongs to the class of functions that satisfy a Lipschitz condition of order $ \alpha < 1 $, then

$$ \| f ( x) - \sigma _ {n} ( f, x) \| _ {c} = \ O \left ( { \frac{1}{n ^ \alpha } } \right ) , $$

that is, in this case the Fejér sum approximates $ f $ at the rate of the best approximating functions of the indicated class. But Fejér sums cannot provide a high rate of approximation: The estimate

$$ \| f ( x) - \sigma _ {n} ( f, x) \| _ {c} = \ o \left ( { \frac{1}{n} } \right ) $$

is valid only for constant functions.

Fejér sums were introduced by L. Fejér [1].

References

[1] L. Fejér, "Untersuchungen über Fouriersche Reihen" Math. Ann. , 58 (1903) pp. 51–69
[2] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[3] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
[4] I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian)
[5] V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)

Comments

See also Fejér summation method.

How to Cite This Entry:
Fejér sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fej%C3%A9r_sum&oldid=15688
This article was adapted from an original article by S.A. Telyakovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article