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.
Let . 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.
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 (1) and (2) over , and (3) over .
- The Fejér kernel
- The Poisson kernel (continuous index)
- The Dirichlet kernel is not a summability kernel, since it fails the second requirement.
Let be a summability kernel, and denote the convolution operation.
- If (continuous functions on ), then in , i.e. uniformly, as .
- 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.