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Given that I have two functions, $f(x)$ and $g(x)$, with $f(x)$ being known, is it possible to recover/deduce the $g(x)$ function if

$$ \int\limits_{ - \infty }^\infty {f(x)g(x)dx = k} $$

where $k$ is known? $f(x)$ can in my case take on a number of different (known) forms, and similar for $k$, but $g(x)$ is fixed, but initially unknown.

Is it possible somehow to get an idea of the shape or functional form of $g(x)$? (The question takes root in a practical problem for me, where I measure $k$ and knows $f(x)$, but would like to know $g(x)$)

Many thanks in advance, and best regards, Bjarke

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  • $\begingroup$ If $g_1(x)=\frac K k g(x)$ then you can get any $K$ value from one non-zero $k$-value, so the only real way for $k$ to determine $g$ is if you are restricting yourself to multiples of some $g_0$ $\endgroup$ Feb 20 '13 at 13:35
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    $\begingroup$ The general way to transform with $f$ is to take $$k(x)=\int f(t)g(x-t)\;dt$$ Then sometimes knowing the entire function $k(x)$ lets you determine the function $g$. See en.wikipedia.org/wiki/Convolution $\endgroup$ Feb 20 '13 at 13:38
  • $\begingroup$ The transformation $$g\to\int_{-\infty}^{\infty}f g dx$$ is not injective. Therefore, you clearly cannot know everything about $g$ if you have only a limited set of test functions $f$. That said, you can know something, but it's hard to say more unless you are more specific about what $f$'s do you have. $\endgroup$
    – yohBS
    Feb 20 '13 at 14:37
  • $\begingroup$ If you are free to choose $f$ then choose an approximation of a Dirac delta function and get point-wise information about $g$. So your $f$ will be simple narrow rectangular blips and you recover the average of function $g$ on the base of that blip. $\endgroup$
    – Maesumi
    Feb 21 '13 at 4:52
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If you know the value of $$ \int f_k g $$ for a suitable choice of known functions $f_k$ you can possibly recover $g$. The sequence $f_k$ must be a base of the Lebesgue space $L^2$. Possible examples are the trigonometric functions which give the fourier coefficients of $g$.

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