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,^[1] but sometimes the definition of an approximation of the identity is taken to be the same as for a summability kernel.

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

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

Note that if ${\displaystyle k_{n}\geq 0}$ for all ${\displaystyle n}$, i.e. ${\displaystyle (k_{n})}$ is a positive summability kernel, then the second requirement follows automatically from the first.

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

${\displaystyle \int _{\delta \leq |t|\leq \pi }|k_{n}(t)|\,dt\to 0}$ as ${\displaystyle n\to \infty }$, for every ${\displaystyle \delta >0}$.

This expresses the fact that the mass concentrates around the origin as ${\displaystyle n}$ increases.

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

• 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.

Let ${\displaystyle (k_{n})}$ be a summability kernel, and ${\displaystyle *}$ denote the convolution operation.

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

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