Take the tour ×  How would I construct Schwartz functions
$f_1^\epsilon$, $f_2^\epsilon$ on $\mathbb{R}$ such that
$f_1^\epsilon(x)\leq\mathbb{1}_{[a,b]}(x)\leq f_2^\epsilon(x)$,
and $f_1^\epsilon\rightarrow\mathbb{1}_{[a,b]}$,
$f_2^\epsilon\rightarrow\mathbb{1}_{[a,b]}$ as
$\epsilon\rightarrow0$,  where $a<b$ are real numbers and
$\mathbb{1}_{[a,b]}$ is the indicator function?  share|improve
this question Hint: $e^{-x^2/2}$ Is this homework? –  Tim Duff
Nov 12 '12 at 6:02 no, it isn't homework –  Kelly Nov 12 '12 at
7:03 You seem to be new to the site - in general, I would
recommend showing the progress you've made on the question so
that your answerer can get a better sense of what you under
stand - helps prevent pedantry and saves time. For this
particular question, it's a fairly lengthy (though standard)
construction, so I'd prefer to get to the part that's tripping
you up. –  Tim Duff Nov 12 '12 at 7:12 add comment  1 Answer
The quickest way is to convolve the characteristic function of
another interv