If you would write the frequency-response in terms of projections through inproducts. How would you do it?

the impulse-response and frequency-response of a signal

Only systems have impulse responses and frequency responses, signals don't. I assume that what you mean here.

A System describes the relation ship between its input signal and its output signal. For an LTI system, that relationship can be captured through either the transfer function or the impulse response. The idea is similar

1. Project the signal onto the some orthogonal basis
2. Run the basis function through the system
3. Assemble output basis function into the output signal. Since it's an LTI system you can simply sum or integrate.

Of course that only makes sense if the bases functions are "easier" to run through the system than some arbitrary signal. For the transfer functions the basis functions are complex exponentials and for the impulse response they are time shifted dirac deltas.

Update

Elaborating on step 2: The basis functions have an index or parameter. For example for a complex exponential, the parameter is frequency. Now you calculate the output of the system with the basis function as a an input using the parameter as a variable. Specifically you calculate the output for a complex exponential with frequency $omega$ as a variable. The result is a function of frequency: That's the transfer function.

So if you have a signal $x(t)$ that can be expressed as the sum of some basis functions b_k(t) like

$$x(t) = sum a_k cdot b_k(t) $$

You can calculate the output as a weighted sum of the outputs of the basis functions.

$$y(t) = Tbegin{Bmatrix} x(t) end{Bmatrix} = Tbegin{Bmatrix} sum a_k cdot b_k(t) end{Bmatrix} = sum a_k cdot Tbegin{Bmatrix} b_k(t) end{Bmatrix} $$