# Definition(s) Let $n$ be a natural number. > [!NOTE] Definition 1 (Complex Trigonometric Polynomials) > A *complex* trigonometric polynomial of degree $2n$ on $[-\pi,\pi]$ is a sum of the form $\sum_{k=-n}^{n} c_{k} e^{ikx}, \quad c_{k}\in \mathbb{C}, x\in [-\pi,\pi].$ > [!Example] Example > Contents > [!NOTE] Definition 2 (Real Trigonometric Polynomials) > A *real* trigonometric polynomial of degree $2n$ on $[-\pi,\pi]$ is a polynomial of the form $a_{0}+\sum_{k=1}^{n}(a_{k}\cos(kx)+b_{k}\sin(kx)), \quad a_{k},b_{k}\in \mathbb{R}, x\in [-\pi,\pi]$ see [[Equivalence of Real and Complex Trigonometric Polynomials]]. # Properties(s) # Application(s) **More examples**: # Reference(s)