The entire problem is like physics-math. Generally we have n cables whose signals are desynchronized and we have m receivers. We know what are waves of voltage at receivers and distances between receivers. We also know that the signals on cables are diffrent only be frequency, phase and amplitude. We want to know the position of each cable with respect to receivers, so I came out with this problem: "I want to decompose a wave into non-sinusoidal waves $u(nx+t)$, where $n$ is the frequency and $t$ is the phase. when I was wondering about this problem I created this equation:$$f(x)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}A(n,t)\cdot u(nx+t) dt dn$$where A(n,t) is the amplitude we want to solve for and we know $u$ and $f$ ($f$ is a function that we want to decompose and $u$ is the function that we want to decompose to). I tried using a inverse of a definite integral function which i called H with this property:$$H^{-1}(a,b,f(x),x) = \int_{a}^{b}f(x)dx$$, but I got nowhere. After simplification this problem becomes: "Given $f,g$ two continuous functions, let's suppose that we know $f(x)$ but we don't know $g(x,n)$. Can you solve for $g(x, n)$ from the following equation $$f(x)=\int_{-\infty}^{\infty} g(x, n) dn$$ or just simplify the integral using $g(x, n)$?" I tried using laplace transform and I arrived at this: $$\mathcal L_n\{g(x,n)\}(s=0)+\mathcal L_{-n}\{g(x,-n)\}(s=0)=f(x)$$ and thanks to Kurt G. I tried fourier transform $$f(x) = \mathcal{F}_{n} \{g(x,n)\} (0)$$ and I don't know what to do with this equation.
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Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Feb 20 '24 at 11:50
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Using $g(x,t)=\frac1{\sqrt{2\pi\sigma^2}}\exp(-t^2/(2\sigma^2))$ gives us plenty of functions whose integral over $t$ is one (one function for every $\sigma$). We cannot determine a single one of those by knowing that integral. – Kurt G. Feb 21 '24 at 17:56
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Besides: if, as you say, we know $f(x)$ what needs to be simplified regarding the integral? – Kurt G. Feb 21 '24 at 18:26
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oh and we know all integrals and derivatives of f(x) – Hubert Kowalewski Feb 21 '24 at 18:43
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1In the counter example I gave $f\equiv 1,.$ We know all its integrals and derivatives. Since $g$ there does not depend on $x$ the derivatives of $f$ won't help much to find $g,.$ Will the integrals help? Think about it. – Kurt G. Feb 21 '24 at 20:12
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g is a function of x and t – Hubert Kowalewski Feb 21 '24 at 20:45
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If a method does not work for $g$ independent of $x$ how could it work when $g$ depends on $x,?$ This counterexample stands and that Laplace transform therefore cannot work wonders. – Kurt G. Feb 22 '24 at 09:08
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Have you checked that Fourier transform? – Kurt G. Feb 22 '24 at 19:52
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Yes i have, but it doesn't give me the result i wanted – Hubert Kowalewski Feb 22 '24 at 20:25
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Even after the edit, this problem still admits trivial counterexamples such as Kurt's – whpowell96 Feb 23 '24 at 16:34
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The Fourier transform of $\sin(x+\frac\pi4)$ is *not* $\frac{\sqrt{2}}{2}\cos(x)+\frac{\sqrt{2}}{2}\sin(x),.$ – Kurt G. Feb 23 '24 at 16:46
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isn't the fourier transform just 2 summations of sin and cos + $a_0$? – Hubert Kowalewski Feb 23 '24 at 16:57
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why fourier transform of $\sin(x+\frac{\pi}{4})$ doesn't equal $\frac{\sqrt{2}}{2}\cos(x)+\frac{\sqrt{2}}{2}\sin(x)$? – Hubert Kowalewski Feb 23 '24 at 17:23
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My second and last link to show you that Fourier transform. I am out of here. – Kurt G. Feb 23 '24 at 18:21
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thx Kurt G. for your help – Hubert Kowalewski Feb 23 '24 at 18:34