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, but sometimes the definition of an approximation of the identity is taken to be the same as for a summability kernel.

Definition
Let $$\mathbb{T}:=\mathbb{R}/\mathbb{Z}$$. A summability kernel is a sequence $$(k_n)$$ in $$L^1(\mathbb{T})$$ that satisfies
 * 1) $$\int_\mathbb{T}k_n(t)\,dt=1$$
 * 2) $$\int_\mathbb{T}|k_n(t)|\,dt\le M$$ (uniformly bounded)
 * 3) $$\int_{\delta\le|t|\le\frac{1}{2}}|k_n(t)|\,dt\to0$$ as $$n\to\infty$$, for every $$\delta>0$$.

Note that if $$k_n\ge0$$ for all $$n$$, i.e. $$(k_n)$$ is a positive summability kernel, then the second requirement follows automatically from the first.

With the more usual convention $$\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$$, the first equation becomes $$\frac{1}{2\pi}\int_\mathbb{T}k_n(t)\,dt=1$$, and the upper limit of integration on the third equation should be extended to $$\pi$$, so that the condition 3 above should be

$$\int_{\delta\le|t|\le\pi}|k_n(t)|\,dt\to0$$ as $$n\to\infty$$, for every $$\delta>0$$.

This expresses the fact that the mass concentrates around the origin as $$n$$ increases.

One can also consider $$\mathbb{R}$$ rather than $$\mathbb{T}$$; then (1) and (2) are integrated over $$\mathbb{R}$$, and (3) over $$|t|>\delta$$.

Examples

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

Convolutions
Let $$(k_n)$$ be a summability kernel, and $$*$$ denote the convolution operation.
 * If $$(k_n),f\in\mathcal{C}(\mathbb{T})$$ (continuous functions on $$\mathbb{T}$$), then $$k_n*f\to f$$ in $$\mathcal{C}(\mathbb{T})$$, i.e. uniformly, as $$n\to\infty$$. In the case of the Fejer kernel this is known as Fejér's theorem.
 * If $$(k_n),f\in L^1(\mathbb{T})$$, then $$k_n*f\to f$$ in $$L^1(\mathbb{T})$$, as $$n\to\infty$$.
 * If $$(k_n)$$ is radially decreasing symmetric and $$f\in L^1(\mathbb{T})$$, then $$k_n*f\to f$$ pointwise a.e., as $$n\to\infty$$. This uses the Hardy–Littlewood maximal function. If $$(k_n)$$ is not radially decreasing symmetric, but the decreasing symmetrization $$\widetilde{k}_n(x):=\sup_{|y|\ge|x|}k_n(y)$$ satisfies $$\sup_{n\in\mathbb{N}}\|\widetilde{k}_n\|_1<\infty$$, then a.e. convergence still holds, using a similar argument.