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


The Fejér kernel is defined as


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



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


The convolution Fn is positive: for of period it satisfies

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

By Young's convolution inequality,

for every

for .

Additionally, if , then


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 uniqueness 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]


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