Summability kernel

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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 $\mathbb {T} :=\mathbb {R} /\mathbb {Z}$ . A summability kernel is a sequence $(k_{n})$ in $L^{1}(\mathbb {T} )$ that satisfies

$\int _{\mathbb {T} }k_{n}(t)\,dt=1$
$\int _{\mathbb {T} }|k_{n}(t)|\,dt\leq M$ (uniformly bounded)
$\int _{\delta \leq |t|\leq {\frac {1}{2}}}|k_{n}(t)|\,dt\to 0$ as $n\to \infty$ , for every $\delta >0$ .

Note that if $k_{n}\geq 0$ 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

$\int _{\delta \leq |t|\leq \pi }|k_{n}(t)|\,dt\to 0$ 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 ${\widetilde {k}}_{n}(x):=\sup _{|y|\geq |x|}k_{n}(y)$ satisfies $\sup _{n\in \mathbb {N} }\|{\widetilde {k}}_{n}\|_{1}<\infty$ , then a.e. convergence still holds, using a similar argument.