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I know that $f(x)$ and $g(x)$ are two continuous functions over $x>0$. If $\int_0^\infty f(x) dx>0$ and $g(x)>0$, $\forall x>0$, under what conditions $\int_0^\infty f(x)g(x) dx>0$? Can we say anything in general?

Salivan
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  • One can easily come up with examples where $\operatorname{sgn}(f(x))=-\operatorname{sgn}(g(x))$ for all $x$ such that the integral of $f$ and $g$ are each positive, but the integral of the product is not. – Michael Burr Mar 27 '17 at 00:38

2 Answers2

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You cannot say anything about the integral of the product. Say, $f(x)=1$, for $x \le 1$, and $f(x)=0$ for $x>1$. If $g(x)=f(x)$, then the integral is positive. But if I choose $g(x)=-10$ for $x\le 1$ and $g(x)=100$ for $x>1$, the integral of $g$ is greater than 0, but the integral of the product is not.

Andrei
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Informally speaking we can think of a functions as a vector. and the integral of the product of two functions as the inner product. Let $f$ be some fixed function, then the space of functions whose inner product with $f$ is positive will be a halfspace, let that space be called $H_f$. I believe that this can be called the dual cone of $f$ wiki. Let $\mathbf{1}$ be the constant one function. What you are asking is that given that $f$ and $g$ lie in $H_{\mathbf{1}}$ will $g$ lie in $H_f$?

As others have noted, and you can see for yourself by just drawing two vectors on a piece of paper, there is not enough information to usefully answer this question, because very many different kinds of conditions can be created. However, one thing that we can say in general is due to Farkas' Lemma. Either $g \in H_f$, or, $\exists$ some function $h$ such that $\int fh >0$ but $\int gh < 0$. So if you can find a certificate function (h) then your job is done.

Pushpendre
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