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 ): Dirichlet summation. As proposed in comments ( Fejér kernel ). Fejér summation.

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 $1$ ): $$s_n(k) = \left(\frac{n}{n+1}\right)^k s_{n-1}(k) + \left(1-\left(\frac{n}{n+1}\right)^k\right) f_n$$

For $k = 2$: mathreadler For $k = \sqrt 2 $ mathreadler2

  • 6
    $\begingroup$ For Fourier series, how about taking the arithmetic mean of the partial sums (replace the Dirichlet kernel with the Fejér kernel)? Since the Fejér kernel is positive, you get no overshoot. $\endgroup$ – Daniel Fischer Aug 21 '15 at 11:25
  • $\begingroup$ By linearity the Fejér kernel should be possible to view as multiplying the frequencies by a triangle where the Dirichlet multiplies with a box function. As the triangle is the convolution of two boxes, I grow curious to try such convolutions of higher order (or other weighted averages). Do you know if that could be useful or any references to more kernels? $\endgroup$ – mathreadler Aug 29 '15 at 16:07
  • $\begingroup$ Try to use frame methods proposed in the following paper: Gibbs phenomenon using tight framelets expans sciencedirect.com/science/article/pii/S100757041730237X $\endgroup$ – Mutaz Mohammad Apr 2 '19 at 7:15

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