# Fejer kernel

The Fejer kernel $F_{n}$ of order $n$ is defined as

 $F_{n}(t)=\frac{1}{n}\sum_{k=0}^{n-1}D_{k}(t),$

where $D_{n}$ is the Dirichlet kernel of order $n$. The Fejer kernel can be written as

 $F_{n}(t)=\frac{1}{n}\left(\frac{\sin\frac{nt}{2}}{\sin\frac{t}{2}}\right)^{2}.$ (1)

Proof: Since

 $D_{n}(t)=\frac{\sin\left(\left(n+\frac{1}{2}\right)t\right)}{\sin\frac{t}{2}}$

we have

 $\sin\frac{t}{2}D_{n}(t)=\sin\left(\left(n+\frac{1}{2}\right)t\right).$

Therefore

 $\displaystyle n\sin^{2}\frac{t}{2}F_{n}(t)$ $\displaystyle=\sum_{k=0}^{n-1}\sin\left(\left(k+\frac{1}{2}\right)t\right)\sin% \frac{t}{2}$ $\displaystyle=\frac{1}{2}\sum_{k=0}^{n-1}(\cos kt-\cos((k+1)t)$ $\displaystyle=\frac{1}{2}(1-\cos nt)$ $\displaystyle=\sin^{2}\frac{nt}{2}.$

From this follows equation (1).

Title Fejer kernel FejerKernel 2013-03-22 14:11:56 2013-03-22 14:11:56 mathwizard (128) mathwizard (128) 8 mathwizard (128) Definition msc 26A30 DiracSequence