The n-th Fourier coefficient is (if I remember correctly) exactly the inner product of f and a function f_n from a fixed orthonormal base.
So you have:
c_n=<f,f_n> (where <.,.> denotes the inner product)
and by Cauchy-Schwarz you have
But maybe I misunderstood your question and you want to put all coefficients in the result of the function. If yes, which sequence space are you working in? l_\infty?