Fourier Transform: Determining Constant in Convolution

[aaaa202]

I have an exercise with a function of the form:

h(t) = f(t)g(t)

and f(t) and g(t) both have discrete Fourier series, which implies that h does too. I want to find the Fourier series of h, so my teacher said I should apply the convolution theorem which would turn the product above into a convolution in the frequency domain.

∑_ωnf(a-ω_n)g(ω_n)

But the problem for me is that it seems arbitrary for me what the constant a should be? How is that determined?

The problem also arises in the case where I have a product as above but I want to find the Fourier transform (i.e. now not series but proper integral transform).
The FT will be a convolution but what determines the constant in the convolution?

 

[jk22]

I suppose you have $$\hat{h}(a)=\sum_{\omega_n}\hat{f}(a-\omega_n)\hat{g}(\omega_n)$$,

so that a is the argument of h.