Is the positive part of the Fourier series of every continuous function , the Fourier series of some other continuous function?

Question

Let $f \in C[-\pi , \pi]$ , let $\hat f (n) : = \dfrac 1{2\pi} \int_{-\pi}^\pi f(t)e^{-int} dt , \forall n \in \mathbb Z$ . Does there exist a continuous function $g : [-\pi , \pi] \to \mathbb C$ such that $\sum _{n=1}^\infty \hat f (n) e^{inx} = \sum_{n=-\infty}^\infty \hat g (n)e^{inx} , \forall x \in [-\pi , \pi]$ ?