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.
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
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.