Hölder summation methods

A collection of methods for summing series of numbers, introduced by O. Hölder [1] as a generalization of the summation method of arithmetical averages (cf. Arithmetical averages, summation method of). The series

$$\sum_{n=0}^\infty a_n$$

is summable by the Hölder method $(H,k)$ to sum $s$ if

$$\lim_{n\to\infty}H_n^k=s,$$

where

$$H_n^0=s_n=\sum_{k=0}^na_k,$$

$$H_n^k=\frac{H_0^{k-1}+\dotsb+H_n^{k-1}}{n+1},$$

$k=1,2,\dotsc$. In particular, $(H,0)$-summability of a series indicates that it converges in the ordinary sense; $(H,1)$ is the method of arithmetical averages. The $(H,k)$-methods are totally regular summation methods for any $k$ and are compatible for all $k$ (cf. Compatibility of summation methods). The power of the method increases with increasing $k$: If a series is summable to a sum $s$ by the method $(H,k)$, it will also be summable to that sum by the method $(H,k')$ for any $k'>k$. For any $k$ the method $(H,k)$ is equipotent and compatible with the Cesàro summation method of the same order $k$ (cf. Cesàro summation methods). If a series is summable by the method $(H,k)$, its terms $a_n$ necessarily satisfy the condition $a_n=o(n^k)$.

References

[1]	O. Hölder, "Grenzwerthe von Reihen an der Konvergenzgrenze" Math. Ann. , 20 (1882) pp. 535–549
[2]	G.H. Hardy, "Divergent series" , Oxford Univ. Press (1949)