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$\begingroup$

Why does this work (for all $a>0$). How can you prove the formula
$$\lim_{L\rightarrow \infty}\int\limits_{-a^2}^{a^2} \frac{f(x) \sin(Lx)}{\pi x} dx=f(0)$$

Does it work for all functions f?

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    hint: $\ \frac {\sin(Lt)}{\pi t}\to \delta(t)\ $ as $L\to\infty$. [eq. (9) of link](http://www.physics.usu.edu/riffe/3750/Lecture%2015.pdf)2012-08-26
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    Why the $a^2$ instead of just $a$?2012-08-26
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    Consult your analysis textbook on "approximations to the identity."2012-08-26
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    It doesn't hold for all functions. Counterexample $$f(x)=\begin{cases}1\quad x=0\\0\quad x\neq 0\end{cases}$$2012-08-26
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    In many text books they consider $f(x)$ to be continuous.2012-08-27
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    So does it hold for all continuous functions?2012-08-27

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