Difference between revisions of "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==== | ====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> | ||
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====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=12552
Bernstein-Rogosinski summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernstein-Rogosinski_summation_method&oldid=12552
This article was adapted from an original article by A.A. Zakharov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article