Is mathematical relation between functions the same as relation between their Fourier series?

Question

For example I have three functions $a(x)$,$b(x)$ and $c(x)$ which are defined on $(-\pi,\pi)$ They have a relation that $a(x)=2b(x)+c(x)$. Assume their Fourier series are $A(x)$ $B(x)$ and $C(x)$. Is the relation among their Fourier series same as functions relations? In other words, is $A(x)=2B(x)+C(x)$ on interval $(-\pi,\pi)$? If yes, how to prove this is true?

Answer 1

Yes, the relation holds on the Fourier transforms, too. The Fourier transform is linear, so it preserves linear relationships.

If you already know that the Fourier transform is linear, you can just cite that fact as a proof. If not, you can prove it straight from the definition using the fact that integrals are linear: rewrite the equation you're trying to show using the integral definitions for $A$, $B$, and $C$, and then use facts you know about integrals, like $\int(2f(x) + g(x))dx = 2\int f(x)dx + \int g(x)dx$.

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(One more thing: remember that the Fourier transform of a function defined on $(-\pi,\pi)$ isn't a function on $(-\pi,\pi)$. It's not relevant to the main thing you're asking here, but you might want to check your notes on that.)