Gronwall summation method

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