When some Fourier coefficients are fixed, can we control the extremals of the function?

Question

Let $n$ be a odd number. Does there exist any $2\pi$-periodic continuous function $f :\mathbb{R}\to \mathbb{R}$ such that the following points simultaneously hold?

1- $-n\lneqq f_{\min}$ (where $f_{\min}$ is the minimum of $f$).
2- If $|k|\leq \frac{n-1}{2}$, the Fourier coefficients $\hat{f}(k)=\sec\frac{k\pi}{n}$.

We recall that the Fourier series of $f$ is just $\sum_{k\in \mathbb{Z}} \hat{f}(k)e^{ikx}$ with Fourier coefficients $\hat{f}(k)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)e^{-ikt}dt$.

p.s. However I am concerned the above question, the following general form can also be considered. For a given a sequence of real numbers $\{a_k\}_{k=-n}^n$ and $\gamma\in \mathbb{R}$, does there exist any $2\pi$-periodic continuous function $f :\mathbb{R}\to \mathbb{R}$ such that the following points simultaneously hold?

1- $\gamma \leq f_{\min}$ (where $f_{\min}$ is the minimum of $f$).
2- If $|k|\leq n$, the Fourier coefficients $\hat{f}(k)=a_k$.

Answer 1

I define the Fourier series as $$f(x) = \tfrac{1}{2}c_0 + \sum_{k=1}^{\infty} c_k \cos kx.$$ The question is a bit unclear on whether a functional form is needed for all $n$, but for a limited range $1\leq n\leq 27$ this works $$f(x)=\tfrac{1}{2}+\sum_{k=1}^{(n-1)/2}\frac{\cos (kx)}{\cos(k\pi/n)}.$$