Dirichlet and Fejer kernels

The Dirichlet and Fejer kernels: Graphed using Maple

In Maple we define the Dirichlet kernel of order m by

d := (m,s) -> sum(cos(k*s),k=-m..m);

[Note that this is the same as the definition of the Dirichlet kernel given in class.] We then define the Fejer kernel of order n by

Kn := s -> 1/(n+1)*sum(d(j,s),j=0..n);

Then, we can plot both simultaneously

n:=10: plot([Kn(s),d(n,s)],s=-Pi..Pi);

to obtain (we hope):

In green is the Dirichlet kernel of order 10, D_{10}.
In red is the Fejer kernel of order 10, K_{10}.

We can see that D_{10} oscillates a lot about 0, which is why the Lebesgue constant satisfies

L_{10} = ||D_{10}||_1 > (1/(2*Pi))*integral(D_{10},-Pi..Pi) = 1.

Note that the lower bound for the Lebesgue constant we found in class is of the order 2/(Pi^2)*(log(10)+1)=0.669... and thus is obviously pretty crude.

(I am grateful to Lee Dickey, for advising me on how to make this page. If he had done it, it would look much less amateurish.)