What Is the Meaning of the Term "Frequency Response'' When We Design a Pitch Detector This is from Klapuri 2003. What is the "frequency response" and why do we model is as multiplied with the vibrating system we want to get the frequency from?
Taking a periodic function with many harmonics, and then filtering it, is a common way of generating (or modeling) periodic signals.

It is how you can model human speech, where S is the sound produced by vocal cords, and H is the frequency response of the cavities formed by larynx, mouth, lungs, etc. And this is how many vocoders analyze and synthesize signals.

Imagine the signal S to be a periodic signal with all harmonics; ideally, S may be an impulse train, so that you have harmonics present at all integer multiples of the fundamental frequency.

Then, H is a filter which will affect each harmonic independently, in order to produce (or model) the desired periodic signal.

Since the variable $k$ corresponds to frequency, we can assume that they are actually DFT coefficients. Applying a filter H to a signal S is a multiplication in frequency (or a convolution in time).