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Given $f(x)$ continuous for all $x\in \mathbb R$, and $f(x)$ nonzero on $\mathbb R$, $0 such that $|f(x)|\leq e^{a|x|}$ for all $x\in \mathbb R$. What conditions should $f$ have so that the integral

$$\int_{-\infty}^{\infty}\bigg|\frac{e^{b|x|}}{f(x)}\bigg|^{2}\,dx$$ be finite?

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    I think you can rewrite the equation to get rid of the exponential stuff. Then the question becomes a real question, and you want to find functions such that the integral of the square of that function is finite. In fact, the "square" part of the integral can be gotten rid of (made redundant). So it all boils down to: what properties does a (>0) function have to have in to ensure it's area (between infinity and minus infinity) is finite. Well, integral of |1/x| is infinity, whereas integral of 1/x^2 is finite. Integral of 1/(xlog(x)) is infinite. Integral of 1/e^x is finite.2012-05-09
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    What things can you say about the functions which give a finite area that you cannot say about functions with infinite area.2012-05-09
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    Edit: if a=b then we are done. If a>b then we need that (eventually) |f(x)| > e^b|x|, and much more restriction. To this end, think about the stuff I said above2012-05-09
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    In fact, that is a highly interesting question. What CAN you say about a function if it's area between 0 and infinity is finite? Of course the same question applied between -infinity and + infinity is the same because you can just reflect the function from the [0,infinity) case in the y-axis.2012-05-09
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    I think there's only so much you can say (in terms of the sandwich theorem and Taylor series) about f, because the "borderline" of functions with infinite area (between -inf and +inf) is not really a solid border of one function, but a range of functions, e.g. 1/x, 1/(xlogx), 1/(xloglogx). And given any one of these functions, we can always find another function. So there is only so much you can say about f. But by the Sandwich theorem, f cannot eventually be greater than any of these functions. e.g. 1/x^2 is eventually less than all of these functions.2012-05-09

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