Can an arbitrary real function be written in terms of quadratures of an arbitrary frequency with time dependent coefficients?

Question

Given a real function $f$, and a frequency $\Omega$, is it the case that there exist two other real functions $I$ and $Q$ such that $f$ can be written as

$$f(t) = I(t) \cos(\Omega t) - Q(t) \sin(\Omega t) \, ?$$

It seems like the convolution theorem should prove that this expression is always possible, at least in the case that the support of the Fourier transform of $f$ is limited to frequencies near $\Omega$, but I have not come up with a bullet-proof argument.

Answer 1

This can be done as follows, given $f$ define $$I(t) = f(t)\cos(\Omega t) \,\,\,\,\,\,\,\, Q(t) = -f(t)\sin(\Omega t)$$ Then $$I(t)\cos(\Omega t) - Q(t)\sin(\Omega t) = f(t)\cos^2(\Omega t) + f(t)\sin^2(\Omega t) = f(t)$$