# 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,^{[1]} 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

- (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 .

## Examples

- The Fejér kernel
- The Poisson kernel (continuous index)
- The Dirichlet kernel is
*not*a summability kernel, since it fails the second requirement.

## Convolutions

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.

## References

- Katznelson, Yitzhak (2004),
*An introduction to Harmonic Analysis*, Cambridge University Press, ISBN 0-521-54359-2