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.

Definition

Let ${\displaystyle \mathbb {T} :=\mathbb {R} /\mathbb {Z} }$. A summability kernel is a sequence $(k_{n})$ 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 $n\to \infty$, for every $\delta >0$.

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

If instead we take the 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 $\pi$.

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

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 ${\displaystyle (k_{n}),f\in {\mathcal {C}}(\mathbb {T} )}$ (continuous functions on $\mathbb {T}$), then ${\displaystyle k_{n}*f\to f}$ in ${\displaystyle {\mathcal {C}}(\mathbb {T} )}$, i.e. uniformly, as $n\to \infty$.
• If ${\displaystyle (k_{n}),f\in L^{1}(\mathbb {T} )}$, then ${\displaystyle k_{n}*f\to f}$ in ${\displaystyle L^{1}(\mathbb {T} )}$, as $n\to \infty$.
• If $(k_{n})$ is radially decreasing symmetric and ${\displaystyle f\in L^{1}(\mathbb {T} )}$, then ${\displaystyle k_{n}*f\to f}$ pointwise a.e., as $n\to \infty$. This uses the Hardy–Littlewood maximal function. If $(k_{n})$ 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.

References

1. http://math.stackexchange.com/questions/114028/approximation-of-the-identity-and-hardy-littlewood-maximal-function

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