Dirichlet kernel and Fejer Kernel and their convergence

Dirichlet kernel and Fejer Kernel

Considered the convergence of Fourier series (or more precisely, the N-th sum SNf) of f (under some differentiability assumptions on f, essentially we need the existence of f0). This problem shows how to improve this convergence to the case without assuming the differentiability of f.

Considering square function:

Using Dirichlet kernel
$D_n(x) = \frac{1}{2\pi} \sum_{k = -n}^{n}{e^{ikx}} = \frac{1}{2\pi} (1+2 \sum_{k = 1}^{n}{cos(kx)} ) = \frac{sin((n+1/2)x)}{2\pi sin(x/2)}$
Let f be a 2π-periodic function. Then the N-th partial sum SNf of its Fourier series is given by the convolution with the Dirichlet kernel :
$(S_N f)(x) =\int_{-\pi}^{\pi} \! f(y)D_N(x-y)dy.$
$(S_N f)(x)= \frac{4}{\pi} \sum_{n = 1}^{\infty}{\frac{sin(2n-1)x}{2n-1}}$
This “average” eliminate the divergent at the boundary of the periodic function and as a result, the zigzag aliasing disappers.( See the result of 20 terms)

Note that the series does not converge when the function bounces between -1 and 1, this is called Gibbs phenomena. Gibbs phenomena

Now considering Fejer Kernel :
Consider the “average” of the partial sum:

$(A_N f)(x)= \frac{1}{N+1} \sum_{m = 0}^{N}{(S_M f)(x)} = \frac{S_0f(x)+S_1f(x)+...+S_Nf(x)}{N+1}$
$(A_N f)(x) =\int_{-\pi}^{\pi} \! f(y)F_N(x-y)dy$ , where $F_n(x) = \frac{sin^2(\frac{N+1}{2}x)}{2\pi (N+1) sin^2(x/2)}$