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=23332