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

1. $\underset{\mathbb{𝕋}}{\int }{k}_n(t)dt=1$
2. $\underset{\mathbb{𝕋}}{\int }|k_n(t)|dt\le M$ (uniformly bounded)
3. $\underset{\delta \le |t|\le \frac{1}{2}}{\int }|k_n(t)|dt\to 0$ as $n\to \mathrm{\infty }$, for every $\delta >0$.

Note that if $k_n\ge 0$ 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 $\mathbb{𝕋}=\mathbb{ℝ}/2\pi \mathbb{\mathbb{Z}}$, the first equation becomes $\frac{1}{2\pi }\underset{\mathbb{𝕋}}{\int }{k}_n(t)dt=1$, and the upper limit of integration on the third equation should be extended to $\pi$, so that the condition 3 above should be

$\underset{\delta \le |t|\le \pi }{\int }|k_n(t)|dt\to 0$ as $n\to \mathrm{\infty }$, 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 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 $(k_n),f\in {𝒞}(\mathbb{𝕋})$ (continuous functions on $\mathbb{𝕋}$), then $k_n*f\to f$ in ${𝒞}(\mathbb{𝕋})$, i.e. uniformly, as $n\to \mathrm{\infty }$. 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 $n\to \mathrm{\infty }$.
• If $(k_n)$ is radially decreasing symmetric and $f\in L^1(\mathbb{𝕋})$, then $k_n*f\to f$ pointwise a.e., as $n\to \mathrm{\infty }$. This uses the Hardy–Littlewood maximal function. If $(k_n)$ is not radially decreasing symmetric, but the decreasing symmetrization ${\tilde{k}}_n(x):=\underset{|y|\ge |x|}{\sup }{k}_n(y)$ satisfies $\underset{n\in \mathbb{\mathbb{N}}}{\sup }\Vert {\tilde{k}}_n{\Vert }_1<\mathrm{\infty }$, then a.e. convergence still holds, using a similar argument.

References

• Katznelson, Yitzhak (2004), An introduction to Harmonic Analysis, Cambridge University Press, ISBN 0-521-54359-2