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 . A summability kernel is a sequence in that satisfies

  1. (uniformly bounded)
  2. as , for every .

Note that if for all , i.e. is a positive summability kernel, then the second requirement follows automatically from the first.

If instead we take the convention , the first equation becomes , and the upper limit of integration on the third equation should be extended to .

We can also consider rather than ; then we integrate (1) and (2) over , and (3) over .

Examples[edit]

Convolutions[edit]

Let be a summability kernel, and denote the convolution operation.

  • If (continuous functions on ), then in , i.e. uniformly, as .
  • If , then in , as .
  • If is radially decreasing symmetric and , then pointwise a.e., as . This uses the Hardy–Littlewood maximal function. If is not radially decreasing symmetric, but the decreasing symmetrization satisfies , then a.e. convergence still holds, using a similar argument.

References[edit]