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

1. [math]\displaystyle{ \int_\mathbb{T}k_n(t)\,dt=1 }[/math]
2. [math]\displaystyle{ \int_\mathbb{T}|k_n(t)|\,dt\le M }[/math] (uniformly bounded)
3. [math]\displaystyle{ \int_{\delta\le|t|\le\frac{1}{2}}|k_n(t)|\,dt\to0 }[/math] as [math]\displaystyle{ n\to\infty }[/math], for every [math]\displaystyle{ \delta\gt 0 }[/math].

Note that if [math]\displaystyle{ k_n\ge0 }[/math] for all [math]\displaystyle{ n }[/math], i.e. [math]\displaystyle{ (k_n) }[/math] is a positive summability kernel, then the second requirement follows automatically from the first.

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

[math]\displaystyle{ \int_{\delta\le|t|\le\pi}|k_n(t)|\,dt\to0 }[/math] as [math]\displaystyle{ n\to\infty }[/math], for every [math]\displaystyle{ \delta\gt 0 }[/math].

This expresses the fact that the mass concentrates around the origin as [math]\displaystyle{ n }[/math] increases.

One can also consider [math]\displaystyle{ \mathbb{R} }[/math] rather than [math]\displaystyle{ \mathbb{T} }[/math]; then (1) and (2) are integrated over [math]\displaystyle{ \mathbb{R} }[/math], and (3) over [math]\displaystyle{ |t|\gt \delta }[/math].

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 [math]\displaystyle{ (k_n) }[/math] be a summability kernel, and [math]\displaystyle{ * }[/math] denote the convolution operation.

• If [math]\displaystyle{ (k_n),f\in\mathcal{C}(\mathbb{T}) }[/math] (continuous functions on [math]\displaystyle{ \mathbb{T} }[/math]), then [math]\displaystyle{ k_n*f\to f }[/math] in [math]\displaystyle{ \mathcal{C}(\mathbb{T}) }[/math], i.e. uniformly, as [math]\displaystyle{ n\to\infty }[/math]. In the case of the Fejer kernel this is known as Fejér's theorem.
• If [math]\displaystyle{ (k_n),f\in L^1(\mathbb{T}) }[/math], then [math]\displaystyle{ k_n*f\to f }[/math] in [math]\displaystyle{ L^1(\mathbb{T}) }[/math], as [math]\displaystyle{ n\to\infty }[/math].
• If [math]\displaystyle{ (k_n) }[/math] is radially decreasing symmetric and [math]\displaystyle{ f\in L^1(\mathbb{T}) }[/math], then [math]\displaystyle{ k_n*f\to f }[/math] pointwise a.e., as [math]\displaystyle{ n\to\infty }[/math]. This uses the Hardy–Littlewood maximal function. If [math]\displaystyle{ (k_n) }[/math] is not radially decreasing symmetric, but the decreasing symmetrization [math]\displaystyle{ \widetilde{k}_n(x):=\sup_{|y|\ge|x|}k_n(y) }[/math] satisfies [math]\displaystyle{ \sup_{n\in\mathbb{N}}\|\widetilde{k}_n\|_1\lt \infty }[/math], 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