Hölder summation methods
A collection of methods for summing series of numbers, introduced by O. Hölder  as a generalization of the summation method of arithmetical averages (cf. Arithmetical averages, summation method of). The series
is summable by the Hölder method $(H,k)$ to sum $s$ if
$k=1,2,\ldots$. 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 $a_n=o(n^k)$.
|||O. Hölder, "Grenzwerthe von Reihen an der Konvergenzgrenze" Math. Ann. , 20 (1882) pp. 535–549|
|||G.H. Hardy, "Divergent series" , Oxford Univ. Press (1949)|
Holder summation methods. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Holder_summation_methods&oldid=23333