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- as, for every.
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 and are integrated over, and over.
Examples
- The Fejér kernel
- The Poisson kernel
- The Landau kernel
- The Dirichlet kernel is not a summability kernel, since it fails the second requirement.
Convolutions
- If , then in, i.e. uniformly, as. In the case of the Fejer kernel this is known as Fejér's theorem.
- If, then in, as.
- If is radially decreasing symmetric and, then pointwise a.e., as. This uses the Hardy–Littlewood maximal function. If is not radially decreasing symmetric, but the decreasing symmetrization satisfies, then a.e. convergence still holds, using a similar argument.