Polynomial transformation on unitaries

Let $F(x)=\sum_{\ell=-L}^{L}c_\ell e^{i\ell x}$ be a trigonometric polynomial satisfying $|F(x)|\leq 1$ for all $x\in\mathbb{R}$. For any $n$-qubit unitary $U$, there exists a quantum circuit with unitary matrix $\mathcal{V}(U)$ such that $$ (\bra{0}\otimes I^{\otimes n})\mathcal{V}(U)(\ket{0}\otimes I^{\otimes n}) = F(U). $$ Here $F(U)=\sum_{\ell=-L}^{L}c_\ell U^\ell$. The circuit uses one ancilla qubit, $2L$ queries to controlled-$U$ or controlled-$U^\dagger$, and $4L+3$ one-qubit rotations.