Fejér kernel

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In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót Fejér (1880–1959).

Plot of several Fejér kernels

Definition[edit]

The Fejér kernel is defined as

where

is the kth order Dirichlet kernel. It can also be written in a closed form as

,

where this expression is defined.[1]

The Fejér kernel can also be expressed as

.

Properties[edit]

The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is with average value of .

Convolution[edit]

The convolution Fn is positive: for of period it satisfies

Since , we have , which is Cesàro summation of Fourier series.

By Young's inequality,

for every

for .

Additionally, if , then

a.e.

Since is finite, , so the result holds for other spaces, as well.

If is continuous, then the convergence is uniform, yielding a proof of the Weierstrass theorem.

  • One consequence of the pointwise a.e. convergence is the uniquess of Fourier coefficients: If with , then a.e. This follows from writing , which depends only on the Fourier coefficients.
  • A second consequence is that if exists a.e., then a.e., since Cesàro means converge to the original sequence limit if it exists.

See also[edit]

References[edit]

  1. ^ Hoffman, Kenneth (1988). Banach Spaces of Analytic Functions. Dover. p. 17. ISBN 0-486-45874-1.