What Is the Meaning of the Term "Frequency Response'' When We Design a Pitch Detector
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).