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Difference between revisions of "Bernstein-Rogosinski summation method"

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One of the methods for summing Fourier series; denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157401.png" />. A trigonometric 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/b/b015/b015740/b0157402.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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is summable by the Bernstein–Rogosinski method at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157403.png" /> to the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157404.png" /> if the following condition is satisfied:
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One of the methods for summing Fourier series; denoted by $  (BR, \alpha _ {n} ) $.  
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A trigonometric 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/b/b015/b015740/b0157405.png" /></td> </tr></table>
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$$ \tag{* }
  
<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/b/b015/b015740/b0157406.png" /></td> </tr></table>
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\frac{a _ {0} }{2}
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+
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\sum _ { k=1 } ^  \infty 
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(a _ {k}  \cos  kx + b _ {k}  \sin  kx )  \equiv \
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\sum _ { k=0 } ^  \infty  A _ {k} (x)
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157407.png" />, is a sequence of numbers, and where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157408.png" /> are the partial sums of the series (*).
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is summable by the Bernstein–Rogosinski method at a point  $  x _ {0} $
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to the value  $  S $
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if the following condition is satisfied:
  
W. Rogosinski [[#References|[1]]] first (1924) considered the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157409.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574010.png" /> is an odd number, and then (1925) the general case. S.N. Bernstein [S.N. Bernshtein] [[#References|[2]]] considered (1930) the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574011.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574012.png" />-method sums the Fourier series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574013.png" /> in the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574015.png" /> at the points of continuity of the function to its value and is one of the [[Regular summation methods|regular summation methods]].
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$$
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\lim\limits _ {n \rightarrow \infty } \
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B _ {n} (x _ {0} ;  \alpha _ {n} ) \equiv \
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\lim\limits _ {n \rightarrow \infty } \
  
The Bernstein–Rogosinski sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574016.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574018.png" />.
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\frac{S _ {n} (x _ {0} + \alpha _ {n} )+S _ {n} (x _ {0} - \alpha _ {n} ) }{2\ }
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=
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$$
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$$
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= \
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\lim\limits _ {n \rightarrow \infty }  \sum _ { k=0 } ^ { n }  A _ {k} (x _ {0} )  \cos  k \alpha _ {n}  = S,
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$$
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 +
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) $.
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The  $  (BR, \alpha _ {n} ) $-
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method sums the Fourier series of a function  $  f \in L[0, 2 \pi ] $
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in the cases  $  \alpha _ {n} = p \pi /2n $
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and  $  \alpha _ {n} = \pi / (2n + 1) $
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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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Beekmann,  K. Zeller,  "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Beekmann,  K. Zeller,  "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>

Latest revision as of 10:58, 29 May 2020


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 [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] [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.

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

[1] W.W. Rogosinski, "Ueber die Abschnitte trigonometischer Reihen" Math. Ann. , 95 (1925) pp. 110–134
[2] S.N. Bernshtein, , Collected works , 1 , Moscow (1952) pp. 37
[3] G.H. Hardy, "Divergent series" , Clarendon Press (1949)

Comments

References

[a1] W. Beekmann, K. Zeller, "Theorie der Limitierungsverfahren" , Springer (1970)
How to Cite This Entry:
Bernstein-Rogosinski summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernstein-Rogosinski_summation_method&oldid=22103
This article was adapted from an original article by A.A. Zakharov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article