Hölder Summation
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Hölder summation is a method for summing
divergent series In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series mus ...
introduced by .


Definition

Given a
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
: a_1+a_2+\cdots, define :H^0_n=a_1+a_2+\cdots+a_n :H^_n=\frac If the limit :\lim_H^k_n 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_H^k_n


See also

*
Cesàro summation In mathematical analysis, Cesàro summation (also known as the Cesàro mean or Cesàro limit) assigns values to some Series (mathematics), infinite sums that are Divergent series, not necessarily convergent in the usual sense. The Cesàro sum ...


References

* * Summability methods {{analysis-stub