Summability kernel

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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[edit]

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

  1. \int_\mathbb{T}k_n(t)\,dt=1
  2. \int_\mathbb{T}|k_n(t)|\,dt\le M (uniformly bounded)
  3. \int_{\delta\le|t|\le\frac{1}{2}}|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.

If instead we take the convention \mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}, the first equation becomes \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 |t|>\delta.

Examples[edit]

Convolutions[edit]

Let (k_n) be a summability kernel, and * denote the convolution operation.

  • If (k_n),f\in\mathcal{C}(\mathbb{T}) (continuous functions on \mathbb{T}), then k_n*f\to f in \mathcal{C}(\mathbb{T}), i.e. uniformly, as n\to\infty.
  • If (k_n),f\in L^1(\mathbb{T}), then k_n*f\to f in L^1(\mathbb{T}), as n\to\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\infty. This uses the Hardy–Littlewood maximal function. If (k_n) is not radially decreasing symmetric, but the decreasing symmetrization \widetilde{k}_n(x):=\sup_{|y|\ge|x|}k_n(y) satisfies \sup_{n\in\mathbb{N}}\|\widetilde{k_n}\|_1<\infty, then a.e. convergence still holds, using a similar argument.

References[edit]