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

|c_n|<=||f||.||f_n||=||f||.

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?