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I am trying to find an example on a sequence of real-valued functions $(f_n(x))_{n\in\mathbb{N}}$ satisfying the following conditions:

i) $f_n$'s are smooth and have compactly supported.

ii) $\int_{\mathbb{R}}f_n(x)dx=0$ for all $n$.

iii) $\int_{\mathbb{R}}|f_n(x)|dx \ge(\ln n)^{-a}$ for all $n$, for some $a>0$.

I have spent a lot of time to search this sequence, but so far I have still not success. Can someone help me this question? Thank you for helping.

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Let $$ f(x)=\begin{cases}x e^{1/(x^2-1)}&\text{if $|x|<1$}\\0&\text{if $|x|\ge 1$}\end{cases}$$ and $f_n(x)=c_nf(x)$ for suitable $c_n$.

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  • $\begingroup$ Yes. I understand your answer. Thank you very much. $\endgroup$ – User3101 Nov 21 '14 at 15:39

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