In mathematics, a summability kernel is a family or sequence of periodic integrable functions satisfying a certain set of properties, listed below.
Certain kernels, such as the Fejér kernel, are particularly useful in Fourier analysis.
Summability kernels are related to approximation of the identity; definitions of an approximation of identity vary,[1] but sometimes the definition of an approximation of the identity is taken to be the same as for a summability kernel.
A summability kernel is a sequence
that satisfies Note that if
is a positive summability kernel, then the second requirement follows automatically from the first.
With the more usual convention
2 π
2 π
, and the upper limit of integration on the third equation should be extended to
π
δ ≤
≤ π
δ > 0
This expresses the fact that the mass concentrates around the origin as
increases.
> δ
be a summability kernel, and
denote the convolution operation.