Talking about Butterworth filters my prof said that a filter is described by its frequency response $H(\omega )$.
What I think is that filter is really described by its transfer function, since there can exist many transfer functions that share the same $H(\omega )$. Was my prof wrong?
A linear time-invariant (LTI) system is indeed completely described by its frequency response. Note that the frequency response is the Fourier transform of the impulse response, which also completely describes the system. So if the Fourier transform of the impulse response exists, then the resulting frequency response must represent a complete characterization of the system.
In many cases both the Fourier transform and the Laplace transform of the impulse response exist, and, consequently, both contain the same information. In these cases, the Laplace transform (transfer function) does not add any information that isn't already represented by the Fourier transform (frequency response).
Was my prof wrong?
No. You're right, you can find multiple systems with the same frequency response, or the same transfer function.
But if you're using something as a (linear, which is often implied) filter, then all you care about is the frequency response (otherwise, you'd probably not be calling the thing "filter"). So, when I define a filter, it's usually sufficient to specify $H(\omega)$.
