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 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.