The Hölder-regularized sum of order 
of a series is obtained by repeatedly
averaging its partial sums. If 
, define
 |
for 
.
The series is Hölder-summable of order 
to 
if 
.
For each positive integer 
, Hölder summability of order 
is equivalent to Cesàro summability
of order 
,
although the intermediate transformed sequences are
different. Ordinary convergence is the order-zero
case.
