Does there exists a non-zero smooth function on Torus, vanishing on an open set such that all its Fourier coefficients are non-zero?

Question

Let us identify the torus $\mathbb{T}$ with $[-1,1).$ Does there exists a non-zero $f\in C^\infty(\mathbb{T})$, vanishing on an open set such that all its Fourier coefficients $a_n$ defined by $$a_n=\int_\mathbb{T}f(x)e^{inx}dx$$ are non-zero for each $n\in \mathbb{Z}?$