Assume that $J$ is the interval $(-\pi,\pi]$. For $k=1,\ldots,2n$, suppose that $\lambda_k$s are real functions on $J$ with $|\lambda_k|=1$, meaning that $\lambda_k(t)$ is either $-1$ or $1$ where $t\in J$.

Let us fix real numbers $b_1,\ldots, b_{2n}$. We define,

$$f:J\to \mathbb{C} : f(t)=\sum_{k=1}^{2n}b_{k}\lambda_k(t)e^{ikt}$$

Q. Is it valid to assert that the cardinality of the set of roots of the function $f$ does not exceed $2n$?