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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 are real numbers and $\mathbb{1}_{[a,b]}$ is the indicator function?

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    Hint: $e^{-x^2/2}$ Is this homework?2012-11-12
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    no, it isn't homework2012-11-12
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    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.2012-11-12

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