---
tag: 1QC5206K07
kind: concept
name: Summability kernel
msc: [42A24, 42A85]
sources:
  - url: https://en.wikipedia.org/wiki/Dirac_delta_function
---

## 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|DCCA2T9GKH]] and the [[Poisson kernel|TCYSWCTYYM]] are summability kernels, and they explain why [[Cesàro means|MA8X1G11DH]] and Abel means of Fourier series converge.
The [[Dirichlet kernel|3JY4BVE4DW]] is not one, which is why ordinary partial sums can fail to converge.

## Explanations

### L1 human

A summability kernel is a family of weights that become a sharper and sharper spike at $0$ while keeping total weight $1$.
Convolving a function with such a weight replaces each value by an average over a shrinking window.
So the averages approach the function itself.
For example, the Fejér kernels $F_N$ are nonnegative bumps of mean $1$ whose width near $0$ is about $1/N$.
They are the periodic case of an [[approximate identity|CDYXTW3PJF]].