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

is summable by the Hölder method to sum if

where

. In particular, -summability of a series indicates that it converges in the ordinary sense; is the method of arithmetical averages. The -methods are totally regular summation methods for any and are compatible for all (cf. Compatibility of summation methods). The power of the method increases with increasing : If a series is summable to a sum by the method , it will also be summable to that sum by the method for any . For any the method is equipotent and compatible with the Cesàro summation method of the same order (cf. Cesàro summation methods). If a series is summable by the method , its terms necessarily satisfy the condition .

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)