Dirichlet vs Fejér

Convolution of a continuous function {f} on the circle {\mathbb T=\mathbb R/\mathbb (2\pi \mathbb Z)} with the Fejér kernel {\displaystyle F_N(x)=\frac{1-\cos (N+1)x}{(N+1)(1-\cos x)}} is guaranteed to produce trigonometric polynomials that converge to {f} uniformly as {N\rightarrow\infty}. For the Dirichlet kernel {\displaystyle D_N(x)=\frac{\sin((N+1/2)x)}{\sin(x/2)}} this is not the case: the sequence may fail to converge to {f} even pointwise. The underlying reason is that {\int_{\mathbb T} |D_N|\rightarrow \infty }, while the Fejér kernel, being positive, has constant {L^1} norm. Does this mean that Fejér’s kernel is to be preferred for approximation purposes?

Let’s compare the performance of both kernels on the function {f(x)=2\pi^2 x^2-x^4}, which is reasonably nice: {f\in C^2(\mathbb T)}. Convolution with {D_2} yields {\displaystyle \frac{1}{2\pi}\int_{-\pi}^{\pi} f(t)D_2(x-t)\,dt = \frac{7\pi^4}{15} -48 \cos x +3 \cos 2x }. The trigonometric polynomial is in blue, the original function in red:

Convolution with D_2
Convolution with D_2

I’d say this is a very good approximation.

Now try the Fejér kernel, also with {N=2}. The polynomial is {\displaystyle \frac{1}{2\pi}\int_{-\pi}^{\pi} f(t)K_2(x-t)\,dt = \frac{7\pi^4}{15} - 32 \cos x + \cos 2x }

Convolution with F_2
Convolution with F_2

This is not good at all.

And even with {N=20} terms the Fejér approximation is not as good as Dirichlet with merely {N=2}.

Convolution with F_20
Convolution with F_20

The performance of {F_{50}} is comparable to that of {D_2}. Of course, a {50}-term approximation is not what one normally wants to use. And it still has visible deviation near the origin, where the function {f} is {C^\infty} smooth:

Convolution with F_50
Convolution with F_50

In contrast, the Dirichlet kernel with {N=4} gives a low-degree polynomial
{\displaystyle \frac{7\pi^4}{15} -48 \cos x +3 \cos 2x -\frac{16}{27}\cos 3x+\frac{3}{16}\cos 4x} that approximates {f} to within the resolution of the plot:

Convolution with D_4
Convolution with D_4

What we have here is the trigonometric version of Biased and unbiased mollification. Convolution with {D_N} amounts to truncation of the Fourier series at index {N}. Therefore, it reproduces the trigonometric polynomials of low degrees precisely. But {F_N} performs soft thresholding: it multiplies the {n}th Fourier coefficient of {f} by {(1-|n|/(N+1))^+}. In particular, it transforms {\cos x} into {(N/(N+1))\cos x}, introducing the error of order {1/N} — a pretty big one. Since this error is built into the kernel, it limits the rate of convergence no matter how smooth the function {f} is. Such is the price that must be paid for positivity.

This reminds me of a parenthetical remark by G. B. Folland in Real Analysis (2nd ed., page 264):

if one wants to approximate a function {f\in C(\mathbb T)} uniformly by trigonometric polynomials, one should not count on partial sums {S_mf} to do the job; the Cesàro means work much better in general.

Right, for ugly “generic” elements of {C(\mathbb T)} the Fejér kernel is a safer option. But for decently behaved functions the Dirichlet kernel wins by a landslide. The function above was {C^2}-smooth; as a final example I take {f(x)=x^2} which is merely Lipschitz on {\mathbb T}. The original function is in red, {f*D_4} is in blue, and {f*F_4} is in green.

Dirichlet wins again
Dirichlet wins again

Added: the Jackson kernel {J_{2N}} is the square of {F_{N}}, normalized. I use {2N} as the index because squaring doubles the degree. Here is how it approximates {f(x)=2\pi^2 x^2-x^4}:

Convolution with J_2
Convolution with J_2
Convolution with J_4
Convolution with J_4
Convolution with J_20
Convolution with J_20

The Jackson kernel performs somewhat better than {F_N}, because the coefficient of {\cos x} is off by {O(1/N^2)}. Still not nearly as good as the non-positive Dirichlet kernel.

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