Difference between revisions of "Hölder summation methods"
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Revision as of 07:54, 26 March 2012
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) 
Hölder summation methods. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=H%C3%B6lder_summation_methods&oldid=22589