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Can some help me prove or disprove the following assertion which I encountered in research? Thanks!

Let $f:\mathbb R\to\mathbb R$ be an analytic function. If for $\forall c > 0$, we can find some $t'>0$ such that

$$\int_{t'}^{t' + 1} {{f^2}(\tau )d\tau } \le c $$ then
$$\mathop {\lim }\limits_{t \to \infty } f(t) = 0.$$

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  • $\begingroup$ What about the function $f(t) = (\sin^2(t) + e^{-t})^t?$ $\endgroup$
    – J. E. Pascoe
    Commented Mar 16, 2015 at 4:16
  • $\begingroup$ This is good, but how do we know that it is analytic? $\endgroup$
    – Lin Letian
    Commented Mar 17, 2015 at 22:56

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It's false. Take for example $$ f(x) =\sum_{n\in {\Bbb Z}} e^{-n(x-n)^2}. $$ Clearly $f(n) \ge 1$ for all integers $n$. Since in intervals of length $1$ the function $f$ is large only in a small neighborhood of an integer, it is easy to see that $\int_x^{x+1} f(t)^2 dt$ tends to zero as $|x| \to \infty$.

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  • $\begingroup$ This is a brilliant counter-example, thanks! $\endgroup$
    – Lin Letian
    Commented Mar 17, 2015 at 22:59

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