Summability kernel

Summary

A summability kernel is a sequence of periodic functions with mean $1$, uniformly bounded integrals of their absolute values, and mass concentrating at $0$. Convolving a function with the terms of a summability kernel gives approximations that converge to it in $L^1$. For a continuous function the convergence is uniform. The Fejér kernel and the Poisson kernel are summability kernels, and they explain why Cesàro means and Abel means of Fourier series converge. The Dirichlet kernel is not one, which is why ordinary partial sums can fail to converge.