View source for Bernstein-Rogosinski summation method

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One of the methods for summing Fourier series; denoted by  $  (BR, \alpha _ {n} ) $. 
A trigonometric series

$$ \tag{* }

\frac{a _ {0} }{2}
 +
\sum _ { k=1 } ^  \infty  
(a _ {k}  \cos  kx + b _ {k}  \sin  kx )  \equiv \ 
\sum _ { k=0 } ^  \infty   A _ {k} (x)
$$

is summable by the Bernstein–Rogosinski method at a point  $  x _ {0} $
to the value  $  S $
if the following condition is satisfied:

$$ 
\lim\limits _ {n \rightarrow \infty } \ 
B _ {n} (x _ {0} ;  \alpha _ {n} )  \equiv \ 
\lim\limits _ {n \rightarrow \infty } \ 

\frac{S _ {n} (x _ {0} + \alpha _ {n} )+S _ {n} (x _ {0} - \alpha _ {n} ) }{2\ }
 =
$$

$$ 
= \ 
\lim\limits _ {n \rightarrow \infty }  \sum _ { k=0 } ^ { n }  A _ {k} (x _ {0} )  \cos  k \alpha _ {n}  =  S,
$$

where  $  \{ \alpha _ {n} \} , \alpha _ {n} > 0, \alpha _ {n} \rightarrow 0 $, 
is a sequence of numbers, and where the  $  S _ {n} (x) $
are the partial sums of the series (*).

W. Rogosinski [[#References|[1]]] first (1924) considered the case  $  \alpha _ {n} = p \pi /2n $, 
where  $  p $
is an odd number, and then (1925) the general case. S.N. Bernstein [S.N. Bernshtein] [[#References|[2]]] considered (1930) the case  $  \alpha _ {n} = \pi / (2n + 1) $. 
The  $  (BR, \alpha _ {n} ) $-
method sums the Fourier series of a function  $  f \in L[0, 2 \pi ] $
in the cases  $  \alpha _ {n} = p \pi /2n $
and  $  \alpha _ {n} = \pi / (2n + 1) $
at the points of continuity of the function to its value and is one of the [[Regular summation methods|regular summation methods]].

The Bernstein–Rogosinski sums  $  B _ {n} (x, \alpha _ {n} ) $
are employed as an approximation procedure. In both cases described above they realize an approximation of the same order as the best approximation for functions of the classes  $  { \mathop{\rm Lip} }  \alpha $
and  $  W  ^ {1} { \mathop{\rm Lip} }  \alpha $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.W. Rogosinski,   "Ueber die Abschnitte trigonometischer Reihen"  ''Math. Ann.'' , '''95'''  (1925)  pp. 110–134</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.N. Bernshtein,  , ''Collected works'' , '''1''' , Moscow  (1952)  pp. 37</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Beekmann,   K. Zeller,   "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>