Summability kernel


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, but sometimes the definition of an approximation of the identity is taken to be the same as for a summability kernel.

Definition

Let. A summability kernel is a sequence in that satisfies
  3. as, for every.
Note that if for all, i.e. is a positive summability kernel, then the second requirement follows automatically from the first.
If instead we take the convention, the first equation becomes, and the upper limit of integration on the third equation should be extended to.
We can also consider rather than ; then we integrate and over, and over.

Examples

Let be a summability kernel, and denote the convolution operation.