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 $\mathbb:=\mathbb/\mathbb$. A summability kernel is a sequence $(k_n)$ in $L^1(\mathbb)$ that satisfies
# $\int_\mathbbk_n(t)\,dt=1$
# $\int_\mathbb, k_n(t), \,dt\le M$ (uniformly bounded)
# $\int_, k_n(t), \,dt\to0$ 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 $\mathbb=\mathbb/2\pi\mathbb$, the first equation becomes $\frac\int_\mathbbk_n(t)\,dt=1$, and the upper limit of integration on the third equation should be extended to $\pi$, so that the condition 3 above should be

$\int_, k_n(t), \,dt\to0$ 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 $\mathbb$ rather than $\mathbb$; then (1) and (2) are integrated over $\mathbb$, 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 $(k_n),f\in\mathcal(\mathbb)$ (continuous functions on $\mathbb$), then $k_n*f\to f$ in $\mathcal(\mathbb)$, i.e. uniformly, as $n\to\infty$. In the case of the Fejer kernel this is known as Fejér's theorem.
* If $(k_n),f\in L^1(\mathbb)$, then $k_n*f\to f$ in $L^1(\mathbb)$, as $n\to\infty$.
* If $(k_n)$ is radially decreasing symmetric and $f\in L^1(\mathbb)$, 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 $\widetilde_n(x):=\sup_k_n(y)$ satisfies $\sup_\, \widetilde_n\, _1<\infty$, then a.e. convergence still holds, using a similar argument.

References

*{{citation
, first=Yitzhak
, last=Katznelson
, authorlink=Yitzhak Katznelson
, title=An introduction to Harmonic Analysis
, year=2004
, publisher=Cambridge University Press
, isbn=0-521-54359-2

Mathematical analysis
Fourier series