View source for Fejér sum

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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 [[#References|[1]]].

====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>

====Comments====
See also [[Fejér summation method|Fejér summation method]].