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

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 $$\hat{f}(k)=a_k$$.

• Do you mean $|k|\le \frac{n-1}2$? Commented Jan 6, 2023 at 9:28
• Yes, thanks and corrected.
– ABB
Commented Jan 6, 2023 at 9:29

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 $$f(x)=\tfrac{1}{2}+\sum_{k=1}^{(n-1)/2}\frac{\cos (kx)}{\cos(k\pi/n)}.$$

• Thanks a lot. If we write, $2\cos(kx)=e^{ikx}+e^{-ikx}$, then it seems that $\hat{f}(k)=\hat{f}(-k)=\frac{1}{2}\sec\frac{k\pi}{n}$ and so the minimum of $f$ is changed.
– ABB
Commented Jan 6, 2023 at 11:23
• I use the definition of Fourier coefficients from Wikipedia: $f(x) = \tfrac{1}{2}c_0 + \sum_{k=1}^{\infty} c_k \cos kx$ --- you may want to specify your definition in the post Commented Jan 6, 2023 at 11:26
• The example that you mentioned is nice and hopefully works for all $n$ (with probably minor change).
– ABB
Commented Jan 6, 2023 at 11:29
• I am extremely interested in the reason of the off vote! Have you ever read it only once? It is logical to write at least one sentence to give negative point.
– ABB
Commented Jan 6, 2023 at 14:32
• @ABB --- voting is anonymous, for what it's worth I did not down vote. Commented Jan 6, 2023 at 14:37