Fejer kernel


The Fejer kernel Fn of order n is defined as

Fn(t)=1nk=0n-1Dk(t),

where Dn is the Dirichlet kernelMathworldPlanetmath of order n. The Fejer kernel can be written as

Fn(t)=1n(sinnt2sint2)2. (1)

Proof: Since

Dn(t)=sin((n+12)t)sint2

we have

sint2Dn(t)=sin((n+12)t).

Therefore

nsin2t2Fn(t) =k=0n-1sin((k+12)t)sint2
=12k=0n-1(coskt-cos((k+1)t)
=12(1-cosnt)
=sin2nt2.

From this follows equation (1).

Figure 1: Graphs of some Fejer kernels
Title Fejer kernel
Canonical name FejerKernel
Date of creation 2013-03-22 14:11:56
Last modified on 2013-03-22 14:11:56
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 8
Author mathwizard (128)
Entry type Definition
Classification msc 26A30
Related topic DiracSequence