I am confused between the transformations between the time-domain and the frequency domain. I have a signal y(t) which is a sum of multiple sinusoids. I band-pass filter this signal to extract one sinusoid and mix this signal with two sinusoids. This is shown in the following figure:
I want to obtain a frequency domain representation of this whole thing: Suppose the signal is Y(s), then the filtered signal is BP(s)Y(s). I then multiply BP(s)Y(s) with the sinusoids $\cos(\omega t)$ and $\sin(\omega t)$, but what does this look like in the frequency domain? I am trying to obtain a single transfer function between Y(s) (the input) and the outputs (the mixed sinusoids).
Convolution in one domain equals point-wise multiplication in the other domain, so multiplication in time domain means convolution in frequency domain. Such convolution in frequency domain does not have a transfer function, because that only holds for multiplication in frequency domain.