Vanishing integral of product of a bounded function with a positive function

Question

I want to know wether one can extract more information from the following equation $$ \int_{x_0}^x f(y) g(y) dy = 0 \,, $$ where $x_0 >0$, $f(x)>0$, and $g(x) \leq const.$ for $x\geq x_0$. In particular, what can we say more about a specific function $g(x)$ which is bounded?

Answer 1

If this is true for a sufficiently large class of test functions $f$, $g$ must be zero (almost everywhere); compactly supported smooth functions are enough, for example. See Fundamental lemma of Calculus of Variations for more details.

Answer 2

If you know that integral is zero for one particular function $f$, and want to know what it tells you about $g$, the answer is "not much". For instance, if $f$ is continuous, then for any high enough frequency $q$, $g(x) = \sin(qx)$ will make the integral almost zero, so as long as $g$ is nonzero somewhere, you can adjust it on a very small domain, by a small amount, to make the integral exactly zero. That says that the functions $g(x) = \sin(1000x)$ and $g(x) = \sin(2019x)$, each slightly altered, will both make the integral zero, but they are pretty wildly different functions. In short: knowing the integral was zero told us almost nothing about $g$.

You could equally well (and almost identically) ask "I have a vector $f$ in $\Bbb R^{100}$. I have a vector $g$ perpendicular to $f$, one that lies within a ball of some radius $C$. What can you tell me about $g$?" The answer is "$g$ lies in the intersection of the hyperplane orthogonal to $f$ with the sphere of radius $C$ around the origin." That doesn't seem to really give you a lot of information.