Hölder summation

In mathematics, Hölder summation is a method for summing divergent series introduced by Hölder (1882).

Definition

Given a series

[math]\displaystyle{ a_1+a_2+\cdots, }[/math]

define

[math]\displaystyle{ H^0_n=a_1+a_2+\cdots+a_n }[/math]

[math]\displaystyle{ H^{k+1}_n=\frac{H^k_1+\cdots+H^k_n}{n} }[/math]

If the limit

[math]\displaystyle{ \lim_{n\rightarrow\infty}H^k_n }[/math]

exists for some k, this is called the Hölder sum, or the (H,k) sum, of the series.

Particularly, since the Cesàro sum of a convergent series always exists, the Hölder sum of a series (that is Hölder summable) can be written in the following form:

[math]\displaystyle{ \lim_{\begin{smallmatrix} n\rightarrow\infty\\ k\rightarrow\infty \end{smallmatrix}}H^k_n }[/math]

See also

• Cesàro summation

References

• Hölder, O. (1882), "Grenzwerthe von Reihen an der Konvergenzgrenze", Math. Ann. 20 (4): 535–549, doi:10.1007/bf01540142, https://zenodo.org/record/1428322 
• Hazewinkel, Michiel, ed. (2001), "Hölder summation methods", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Hölder_summation_methods