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If I have, for example, two finite lists of sine waves and all the sine waves in each of these lists are summed to create 2 new sine waves a and b, then is there any way I can figure out whether or not the 2 lists of sine waves contained any common sine waves that appeared in both lists without the underlying data (the sine waves in the lists) and only the sine waves a and b?

For Example, if I have two lists containing a finite number of sine waves: (sin(x), sin(2x), sin(4x), sin(7x)) and (sin(3x), sin(6x), sin(5x), sin(7x)), and all the sine waves in each list are summed to create two new sine waves a and b: sin(1x) + sin(2x) + sin(4x) + sin(7x) = sine wave a, and sin(3x) + sin(6x) + sin(5x) + sin(7x) = sin wave b, then is is possible to figure out that sin(7x) appears in both lists without knowing what the sine waves in the lists were and only knowing the sine waves a and b?

CCS
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    It's nor clear what you mean by "sine wave". I thought you were referring to a function of time, i.e., of the form $\sin(2 \pi f t)$ with $f$ a (constant) frequency and $t$ representing time, but notation like "sin(1)" in your last paragraph suggests something different. If my understanding is indeed correct, you can always apply Fourier transform to any function and you will find its representation as a sum of sine waves. So you can actually find all sine waves of each list from the fourier transform of each sum, and, in turn if they have common terms. – Stelios May 08 '21 at 01:59
  • Sorry, I forgot to make it more clear what i mean. By lists of sine waves, i meant something like this: (sin(x), sin(7x), sin(3x), sin(4x)). – CCS May 08 '21 at 02:05
  • I think the functions $\sin(nx)$ are orthogonal with respect to the $L^2$ inner product $\langle f, g \rangle = \int_0^\pi f(x) g(x) , dx$. If your combined wave $a$ is a linear combination of these functions, then you can actually recover the coefficient corresponding to $\sin(nx)$ by integrating $\int_0^\pi a(x) \sin(n x),dx$ (there might be some scaling by $\sqrt{\int_0^\pi |a(x)|^2 , dx}$ since the orthogonal basis isn't orthonormal... I'm just sketching the broad idea). For more details, look up "Fourier series" etc. – angryavian May 08 '21 at 02:16

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