Let :=\mathbb {R} /\mathbb {Z} }
. A summability kernel is a sequence in that satisfies
- (uniformly bounded)
- as , for every .
Note that if for all , i.e. is a positive summability kernel, then the second requirement follows automatically from the first.
With the more usual convention , the first equation becomes , and the upper limit of integration on the third equation should be extended to , so that the condition 3 above should be
as , for every .
This expresses the fact that the mass concentrates around the origin as increases.
One can also consider rather than ; then (1) and (2) are integrated over , and (3) over .