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Difference between revisions of "Gronwall summation method"

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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T.H. Gronwall,   "Summation of series and conformal mapping"  ''Ann. of Math.'' , '''33''' :  1  (1932)  pp. 101–117</TD></TR></table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  T.H. Gronwall, "Summation of series and conformal mapping"  ''Ann. of Math.'' , '''33''' :  1  (1932)  pp. 101–117 {{ZBL|0003.05501}}</TD></TR>
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[[Category:Sequences, series, summability]]

Latest revision as of 14:34, 15 June 2024

A method for summing series of numbers or functions, defined by specifying two functions $f$ and $g$ satisfying certain conditions. A series $\sum_{n=0}^\infty u_n$ can be summed by the Gronwall method $(f,g)$ to a sum $s$ if

$$\lim_{n\to\infty}U_n=s,$$

where $U_n$, $n=0,1,\dots,$ is defined by the expansion

$$\sum_{n=0}^\infty u_nz^n=\frac{1}{g(w)}\sum_{n=0}^\infty b_nU_nw^n,$$

$$z=f(w),\quad g(w)=\sum_{n=0}^\infty b_nw^n.$$

The method was introduced by T.H. Gronwall [1] as a generalization of the de la Vallée-Poussin summation method, to which it is converted by

$$z=f(w)=\frac{1-\sqrt{1-w}}{1+\sqrt{1+w}},\quad w=\frac{4z}{(1+z)^2},\quad g(w)=\frac{1}{\sqrt{1-w}}.$$

If

$$f(w)=w,\quad g(w)=(1-w)^{-k-1},$$

then the Gronwall summation method becomes one of the Cesàro summation methods.

References

[1] T.H. Gronwall, "Summation of series and conformal mapping" Ann. of Math. , 33 : 1 (1932) pp. 101–117 Zbl 0003.05501
How to Cite This Entry:
Gronwall summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gronwall_summation_method&oldid=33495
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article