All smooth periodic functions with bounded derivatives I am currently working on Fourier series and the following questions arose in this context:
Which smooth $2\pi$-periodic functions (smooth on $\mathbb{R}$) exist, which themselves ($n = 0$) and for every $n$-th derivative and all $x\in \mathbb{R}$ satisfy the inequality $-1\leq f^{(n)}(x) \leq1$?
These functions can all be represented in a Fourier series. Obviously this is fulfilled by sin and cos. For trigonometric polynomials $f(x)=a_0+\sum_{k=0}^d (a_k \cos(kx)+ b_k \sin(kx))$  with degree $d>2$ it seems like the inequality cannot be fullfilled since the coefficients are exponentially growing, when we take derivatives. So what about other fourier series? I know that for example $\sin$ can be represented as fourier series of $\cos$, so this series would fullfill the inequality. Are there any other series, which are not converging to $\sin$ or $\cos$ or a shifted version of them? And how could I proof that?

The notation will be simpler with the complex version of Fourier series:
Say $$f(x)=\sum_kc_ke^{ikx}.$$Then for each $k_0$ we have$$|k_0|^{2n}|c_{k_0}|^2
\le\sum_k|k|^{2n}|c_k|^2=||f^{(n)}||_2^2\le||f^{(n)}||_\infty^2\le1.$$
If $|k_0|>1$ then letting $n\to\infty$ shows that $c_{k_0}=0$. So $$f(x)=\sum_{|k|\le1}c_ke^{ikx}=a\cos(x)+b\sin(x)+c.$$Now you just have to figure out what $a,b,c$ actually give $||f^{(n)}||_\infty\le 1$. (Start with some trig, rewriting $a\cos(x)+b\sin(x)=A\cos(x+\phi)$; the condition I get is $(a^2+b^2)^{1/2}+|c|\le1$.)
