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)