I have tried to give only the intuitive part of my question and haven't included many specific details. Please help me frame it more precisely. I have inluded the symbol (*) where I need more details.
I was really delighted when I first saw the " Munsz theorem " related to density of a subset of polynomials in the space of continuous functions. When I was reading about Fourier Series, I saw that we are basically trying to approximate functions (*) using dilates of the $sin$ function.$( sin x ,sin 2x,sin 3x,...)$. The periods of the functions are ($ 2 \pi/1, 2\pi/2,2\pi/3,2\pi/4,...) $.
Now, can we ask the same question as in the case of Munsz theorem i.e., can we write a $fourier series$ (*) of the function to be approximated using a subset of the set of dilates of the $sin$ function ? Are there any results regarding this question available already ?
Thank You.