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Bernstein-Rogosinski summation method

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One of the methods for summing Fourier series; denoted by . A trigonometric series

(*)

is summable by the Bernstein–Rogosinski method at a point to the value if the following condition is satisfied:

where , is a sequence of numbers, and where the are the partial sums of the series (*).

W. Rogosinski [1] first (1924) considered the case , where is an odd number, and then (1925) the general case. S.N. Bernstein [S.N. Bernshtein] [2] considered (1930) the case . The -method sums the Fourier series of a function in the cases and at the points of continuity of the function to its value and is one of the regular summation methods.

The Bernstein–Rogosinski sums 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 and .

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://www.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