Trigonometric polynomial

Remarks

By Euler's formula the same functions are the sums $a_0 + \sum_{n=1}^{N} (a_n \cos nx + b_n \sin nx)$ with complex $a_n, b_n$, built from sine and cosine. The two forms are related by $a_0 = c_0$, $a_n = c_n + c_{-n}$ and $b_n = i(c_n - c_{-n})$ for $n \ge 1$. $T$ is real-valued exactly when $c_{-n} = \overline{c_n}$ for all $n$, that is, when all $a_n$ and $b_n$ are real. Example: $\cos^2 x = \frac{1}{4} e^{-2ix} + \frac{1}{2} + \frac{1}{4} e^{2ix}$, so $\cos^2 x$ is a trigonometric polynomial of degree $2$. Non-example: $\sin(x/2)$ is not a trigonometric polynomial, since it has period $4\pi$ and not $2\pi$, while every trigonometric polynomial is $2\pi$-periodic. The coefficients are unique: $c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} T(x) e^{-inx} \, dx$.