I'm looking for a systematic way to approximate the sign function

$$sgn(x)=\begin{cases}1&\text{ if }x\geq 0\\-1&\text{ if }x<0,\end{cases}$$

by functions $f$ whose (distributional) Fourier transforms $\hat{f}$ are compactly supported.

More precisely, let us fix some $\epsilon$ small. Then I am interested in odd, continuous $f$ such that:

  • $\text{supp}{\hat{f}}\subseteq[-\pi,\pi];$
  • there exists some $a_f>0$ such that $|f^2(x)-1|<\epsilon$ whenever $|x|\geq a_f$.

Goal: I am looking for a "best" $f$ in the sense $a_f$ is as small as possible.

I figured that one way to do this would to be to look at convolutions $f=a*sgn$ where $a$ is a function that satisfies $\int_\mathbb{R} a(s)\,ds=1$ and $\text{supp }{\hat{a}}\subseteq[-\pi,\pi]$. For example, we could take

  1. $a_1(s)=\dfrac{\sin(\pi s)}{\pi s}$; or
  2. $a_2(s)=\left(\dfrac{\sin(\pi s)}{\pi s}\right)^2,$

where the Fourier transforms of $a_1$ and $a_2$ are a rectangular and a triangle pulse respectively.

Doing some numerical experiments, I find that for the choice $\epsilon=0.05$, $a_2$ is "better" than $a_1$ by the above criterion.

Question: Is there a more systematic way to find better (or a best) $f$? References that deal with this type of problem would also be appreciated.

  • $\begingroup$ I assume you want a family of functions parameterized by some parameter k. How about $$y=\frac{e^{2k}x}{\sqrt{e^kx^2+1}}$$ $\endgroup$ Nov 2 '20 at 15:05
  • $\begingroup$ You should maybe work with the Heaviside function, a close cousin of sign function... $\endgroup$
    – Jean Marie
    Nov 2 '20 at 15:07
  • $\begingroup$ Have a look at the dsp "branch" of Stack Exchange, for example see this $\endgroup$
    – Jean Marie
    Nov 2 '20 at 15:13
  • $\begingroup$ @JeanMarie Ah thanks, I'll take a a look! $\endgroup$
    – geometricK
    Nov 2 '20 at 15:20

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