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Difference between revisions of "Gronwall summation method"

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A method for summing series of numbers or functions, defined by specifying two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045140/g0451401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045140/g0451402.png" /> satisfying certain conditions. A series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045140/g0451403.png" /> can be summed by the Gronwall method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045140/g0451404.png" /> to a sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045140/g0451405.png" /> if
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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
  
<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/g/g045/g045140/g0451406.png" /></td> </tr></table>
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$$\lim_{n\to\infty}U_n=s,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045140/g0451407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045140/g0451408.png" /> is defined by the expansion
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where $U_n$, $n=0,1,\dots,$ is defined by the expansion
  
<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/g/g045/g045140/g0451409.png" /></td> </tr></table>
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$$\sum_{n=0}^\infty u_nz^n=\frac{1}{g(w)}\sum_{n=0}^\infty b_nU_nw^n,$$
  
<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/g/g045/g045140/g04514010.png" /></td> </tr></table>
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$$z=f(w),\quad g(w)=\sum_{n=0}^\infty b_nw^n.$$
  
 
The method was introduced by T.H. Gronwall [[#References|[1]]] as a generalization of the [[De la Vallée-Poussin summation method|de la Vallée-Poussin summation method]], to which it is converted by
 
The method was introduced by T.H. Gronwall [[#References|[1]]] as a generalization of the [[De la Vallée-Poussin summation method|de la Vallée-Poussin summation method]], to which it is converted by
  
<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/g/g045/g045140/g04514011.png" /></td> </tr></table>
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$$z=f(w)=\frac{1-\sqrt{1-w}}{1+\sqrt{1+w}},\quad w=\frac{4z}{(1+z)^2},\quad g(w)=\frac{1}{\sqrt{1-w}}.$$
  
 
If
 
If
  
<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/g/g045/g045140/g04514012.png" /></td> </tr></table>
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$$f(w)=w,\quad g(w)=(1-w)^{-k-1},$$
  
 
then the Gronwall summation method becomes one of the [[Cesàro summation methods|Cesàro summation methods]].
 
then the Gronwall summation method becomes one of the [[Cesàro summation methods|Cesàro summation methods]].

Revision as of 20:09, 4 October 2014

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
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