In the frequency domain (using variable \omega for frequency), say the input-output relationship of a linear time-invariant (LTI) filter for any frequency \omega \in \mathbb{R} is given by the product
Y(\omega) = H(\omega) X(\omega)
where Y(\omega) is the output signal, X(\omega) is the input signal, and H(\omega) is the frequency response or Fourier domain transfer function of the filter.
The convolution theorem says that if x, y \in L^1 † then
y(t) = \mathcal{F}^{-1}\{H(\omega)\} \ast x(t)
where \mathcal{F}^{-1} is the inverse Fourier transform and \ast is the convolution operator, making it possible to compute a time-domain output of the filter for any L^1 input x.
This all means basically that you can use lots of curves H(\omega) as a frequency response, with the condition for inverse-Fourier-transformability being something like “absolute area under the curve is finite”. So certainly not “any curve” generally speaking, but in practice you are most likely working with digital signals with some finite sampling density and precision. That changes all the math some but the core idea is basically the same: take an inverse Fourier transform of the desired frequency response, convolve it with your input. When working with digital signals this is generally called finite impulse response filter design.
If one is equating “sounds” with, say, "bounded continuous functions \mathbb{R} \to \mathbb{R}", then maybe it is interesting to explore what classes of these functions are not representable in such-and-such basis, e.g. Fourier, Laplace, various wavelets, … Or maybe “bounded continuous” is already too restrictive?
† A function f: \mathbb{R} \to \mathbb{R} belongs to the space L^1 if \int_{-\infty}^{\infty} |f(t)| dt < \infty.
