Dirichlet, Fejér and more

The Dirichlet kernel $$D_N(x)=\sum _{n=-N}^{N}e(nx)$$ has a pivotal importance when studying Fourier series. In some sense it is an approximation of the 1-periodic Dirac delta when $N\to \mathrm{\infty }$. The slow decay is an issue when one enters into convergence problems.

A more regular version is the Fejér kernel, which is the Cesàro sum of the Dirichlet kernel except for shifting $N$ into $N-1$, $$F_N(x)=\sum _{n=-N}^N(1-\frac{|n|}{N})\exp (2\pi inx)$$ The quickest decay of this function at the cost of having thicker peaks (this also appears in digital windows) allows to prove a nice theorem (Fejér's theorem): If $f$ is 1-periodic and continuous having $\{a_m{\}}_{m\in \mathbb{Z}}$ as Fourier coefficients, then $$f(x)=\underset{N\to \mathrm{\infty }}{lim}\sum _{m=-N}^N(1-\frac{|m|}{N})a_m\exp (2\pi imx)\text{uniformly in }x.$$

One may consider smoother versions, so to speak, of the Dirichlet kernel taking higher Cesàro sums. The natural object for $k$ a nonnegative integer is $$C_{N,k}(x)=\sum _{n=-(N-k)}^{N-k}(\genfrac{}{}{0ex}{}{N-k}{|n|}){(\genfrac{}{}{0ex}{}{N}{|n|})}^{-1}\exp (2\pi inx).$$ Note that $$D_N(x)=C_{N,0}(x)\text{and}{F}_N(x)=C_{N,1}(x).$$

The following plots drawn with the code below show the cases $k=0,1,2$ for some small values of $N$.

$N=4,$   $k=0$	$N=4,$   $k=1$	$N=4,$   $k=2$

$N=8,$   $k=0$	$N=8,$   $k=1$	$N=8,$   $k=2$

$N=16,$   $k=0$	$N=16,$   $k=1$	$N=16,$   $k=2$

$N=32,$   $k=0$	$N=32,$   $k=1$	$N=32,$   $k=2$

It is interesting to observe the vertical scale. When $k$ grows the peaks are shorter and wider. Note that ${\int }_{-1/2}^{1/2}{C}_{N,k}=1$ as it must be if we want to mimic a Dirac delta.

The code