View source for Bernstein-Rogosinski summation method

One of the methods for summing Fourier series; denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157401.png" />. A trigonometric series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157402.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

is summable by the Bernstein–Rogosinski method at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157403.png" /> to the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157404.png" /> if the following condition is satisfied:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157405.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157406.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157407.png" />, is a sequence of numbers, and where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157408.png" /> are the partial sums of the series (*).

W. Rogosinski [[#References|[1]]] first (1924) considered the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b0157409.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574010.png" /> is an odd number, and then (1925) the general case. S.N. Bernstein [S.N. Bernshtein] [[#References|[2]]] considered (1930) the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574011.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574012.png" />-method sums the Fourier series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574013.png" /> in the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574015.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574016.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015740/b01574018.png" />.

====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>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Beekmann,   K. Zeller,   "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>