Difference between revisions of "Gronwall summation method"
From Encyclopedia of Mathematics
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| − | A method for summing series of numbers or functions, defined by specifying two functions | + | {{TEX|done}} |
| + | 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 | + | 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 [[#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 | ||
| − | + | $$z=f(w)=\frac{1-\sqrt{1-w}}{1+\sqrt{1+w}},\quad w=\frac{4z}{(1+z)^2},\quad g(w)=\frac{1}{\sqrt{1-w}}.$$ | |
If | If | ||
| − | + | $$f(w)=w,\quad g(w)=(1-w)^{-k-1},$$ | |
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]]. | ||
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====References==== | ====References==== | ||
<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> | <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> | ||
| + | |||
| + | [[Category:Sequences, series, summability]] | ||
Latest revision as of 21:48, 18 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=12127
Gronwall summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gronwall_summation_method&oldid=12127
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article