Bernstein-Rogosinski summation method

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)

References

[a1]	W. Beekmann, K. Zeller, "Theorie der Limitierungsverfahren" , Springer (1970)