Let $f$ be a real trigonometric polynomial of $d$ variables that is bounded $|f|\le 1$. Further assume that the maximum degree for each variable is 1. Then we can write it as the sum of monomials, each of which is a product of sines and cosines of the variables of frequency at most 1. In slightly nonstandard notation$$f(\theta) = \sum_{k \in \{0, \pm 1\}^d} c_{k} \prod_{i=1}^d t_{k_i}(\theta_i)$$where $t_0(\theta) = 1, \,t_1(\theta) = \cos(\theta), \,t_{-1}(\theta) = \sin(\theta)$. The coefficients $c_k$ are real and we place the further restriction that$$c_k \in \{0, \pm 1\}.$$For instance, with $d=2$ one such function is $f(\theta) = \cos(\theta_1)\cos(\theta_2) - \sin(\theta_1)\sin(\theta_2)$.
I have a general question and a more specific question:
- What conditions coming from boundedness are there on the coefficients? I could derive some simple conditions by considering the function on the points $\{0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\}^d$, however these are provably not exhaustive.
- Using these general conditions or otherwise, can it be shown that the function $f_l$ obtained by keeping only the terms with $\|k\|_1 \le l$ is also bounded $|f_l|\le 1$?