What is your favourite method which would help reduce the Gibbs phenomenon in Fourier Series and Fourier Transforms? This could mean pre-processing or post-processing or altering the transform.
With the Gibbs phenomenon I mean the "overshoot" close to a step discontinuity like in the image below.
Ordinary Fourier transform ( Dirichlet kernel )
v.s. proposed in comments ( Fejér kernel ) :
Own work:
Let $f_n$ be the $n$'th Dirichlet kernel (multiplying with a box function which is $1$ for the $n$ lowest frequencies and $0$ otherwise).
Inspired by the Fejér kernel above, realizing we can write it recursively as:
$$s_n = \frac{n}{n+1} s_{n-1} + \frac{1}{n+1} f_n = \frac{n}{n+1} s_{n-1} + \left(1-\frac{n}{n+1}\right) f_n$$ we introduce a family of weighted averages ( which obviously will sum to $1$ ): $$s_n(k) = \left(\frac{n}{n+1}\right)^k s_{n-1}(k) + \left(1-\left(\frac{n}{n+1}\right)^k\right) f_n$$
$$\text{For } k = 2 $$
$$\text{For } k = \sqrt 2 : $$
