A regular approximation of characteristic functions

Is sometimes useful to have a regular function close to the characteristic function of an interval.

More precisely, we want a $C^{\mathrm{\infty }}$ function ${\chi }_{a,b,\delta }:\mathbb{R}\to \mathbb{R}$ verifying:

• $\text{supp}({\chi }_{a,b,\delta })=(a,b)$
• ${\chi }_{a,b,\delta }(x)=1$ for every $x\in [a+\delta ,b-\delta ]$.

An example of such a function is the following one:

$${\chi }_{a,b,\delta }(x)=\{\begin{array}{ll}0& \text{if }x\notin (a,b)\\ 1& \text{if }x\in [a+\delta ,b-\delta ]\\ \frac{1}{2}+\frac{1}{2}\tanh (\tan (-\frac{\pi }{2}+\frac{x-a}{\delta }\pi ))& \text{if }x\in (a,a+\delta )\\ \frac{1}{2}+\frac{1}{2}\tanh (\tan (\frac{\pi }{2}-\frac{x-b+\delta }{\delta }\pi ))& \text{if }x\in (b-\delta ,b)\end{array}$$

Bellow is a python implementation of this function

import numpy as np

def cutoff(x, a, b, d):
    y = 1.0 * (x >= a + d) * (x <= b - d)
    i = np.argwhere((x > a) * (x < a + d))
    y[i] = y[i] + 0.5 + 0.5*np.tanh(np.tan(-np.pi/2 + (x[i] - a)/d*np.pi))
    i = np.argwhere((x > b-d)*(x < b))
    y[i] = y[i] + 0.5 + 0.5*np.tanh(np.tan(np.pi/2 - (x[i] - b + d)/d*np.pi))
    return y

and its graphic representation for $a=0.2$, $b=0.8$ and $\delta =0.2$.