Hölder summation

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

Definition

Given a series

$a_{1} a_{2} \cdots ,$

define

$H_{n}^{0}=a_{1} a_{2} \cdots a_{n}$

$H_{n}^{k 1}={\frac {H_{1}^{k} \cdots H_{n}^{k}}{n}}$

If the limit

$\lim _{n\rightarrow \infty }H_{n}^{k}$

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:

$\lim _{\begin{smallmatrix}n\rightarrow \infty \\k\rightarrow \infty \end{smallmatrix}}H_{n}^{k}$

See also

• Cesàro summation

References

• Hölder, O. (1882), "Grenzwerthe von Reihen an der Konvergenzgrenze", Math. Ann., 20 (4): 535–549, doi:10.1007/bf01540142, S2CID 124308783535-549&rft.date=1882&rft_id=info:doi/10.1007/bf01540142&rft_id=https://api.semanticscholar.org/CorpusID:124308783#id-name=S2CID&rft.aulast=Hölder&rft.aufirst=O.&rft_id=https://zenodo.org/record/1428322&rfr_id=info:sid/en.wikipedia.org:Hölder summation" class="Z3988">
• "Hölder summation methods", Encyclopedia of Mathematics, EMS Press, 2001 [1994]