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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 [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=46023
This article was adapted from an original article by A.A. Zakharov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article