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  "body": "\n## Definition\n\nLet $N \\ge 0$ be an integer.\nA *trigonometric polynomial* of degree at most $N$ is a function $T \\colon \\R \\to \\C$ with\n$$T(x) = \\sum_{n=-N}^{N} c_n e^{inx},$$\nfor complex numbers $c_{-N}, \\dots, c_N$, where $e^{inx}$ is the [[complex exponential|07GJV36896]].\nIts *degree* is the largest $\\abs{n}$ with $c_n \\ne 0$.\n\n## Remarks\n\nBy 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|08WFZP6P89]].\nThe two forms are related by $a_0 = c_0$, $a_n = c_n + c_{-n}$ and $b_n = i(c_n - c_{-n})$ for $n \\ge 1$.\n$T$ is real-valued exactly when $c_{-n} = \\overline{c_n}$ for all $n$, that is, when all $a_n$ and $b_n$ are real.\nExample: $\\cos^2 x = \\frac{1}{4} e^{-2ix} + \\frac{1}{2} + \\frac{1}{4} e^{2ix}$, so $\\cos^2 x$ is a trigonometric polynomial of degree $2$.\nNon-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.\nThe coefficients are unique: $c_n = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} T(x) e^{-inx} \\, dx$.\n",
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