I am looking for a function $f\in L_{\infty}[-\pi,\pi]$ which is $2\pi$ periodic and
$$||f||_{\infty}\leq 1$$
and
$$||\sum_{k=-N}^{N} \hat{f}(k) e^{ikx}||_{\infty} \gt 1$$
where $\hat{f}(k)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-ikx}dx$ is Fourier coefficient.
Does such a function exist?