Let's take the Fourier transform of Faraday's law of induction
$$ \nabla \times E = - \partial_t B $$
$$ \mathcal{F}[\nabla \times E]\ = \mathcal{F}[- \partial_t B] $$
$$ \nabla \times \mathcal{F}[E]\ = - \mathcal{F}[\partial_t B] $$
Using integration by parts $u = e^{- i \omega t}$, $du = - i \omega e^{i \omega t}$, $dv = \partial_t B$, $v = B$, so the right side becomes
$$ e^{- i \omega t} B \ \big|_{-\infty}^\infty + i \omega \int_{-\infty}^\infty e^{- i \omega t} B$$
Now I know from other sources that the left term is $0$, but why? We are solving for B so how do we know its characteristics yet?
Also, how do we know that $\mathcal{F}[E]$ and $\mathcal{F}[B]$ exist? Certainly plane waves solutions satisfy Maxwell's equations but do not have the correct boundary condition to make the left term $0$. Does this show that the Fourier transform does not give a general solution like the method of separation of variables does?
More generally how do we know any integral transform of an unknown function exists? Take for example the Laplace transform of $p(x)y'' + q(x)y' + r(x)y = f(x)$. I know the Laplace transform can be applied when each coefficient is bounded by some exponential function. But that only guarantees that the Laplace transform of each coefficient exists. Sure we can go through the problem and solve for $\mathcal{L}[y] =$ some function of $s$, so we know it exists that way. But is there any way to guarantee that $\mathcal{L}[y]$ exists before we solve the problem?