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  "body": "\n## Definition\n\n*Fourier analysis* is the part of [[harmonic analysis|7QNRC9SJVV]] that studies the decomposition of functions into [[complex exponentials|07GJV36896]] and the reconstruction of functions from these components.\nIt covers the following transforms:\n\n1. the [[Fourier series|G12KXDHPPS]] of a $2\\pi$-periodic function, with components $e^{inx}$ for $n \\in \\Z$;\n2. the [[Fourier transform|DPVM34ADZV]] of a function on $\\R^n$, with components $e^{2\\pi i x \\cdot \\xi}$ for $\\xi \\in \\R^n$;\n3. the discrete Fourier transform of a function on $\\Z / N\\Z$, with components $e^{2\\pi i jk/N}$ for $k \\in \\Z / N\\Z$.\n\n## Remarks\n\nExample: the $2\\pi$-periodic function $f(x) = \\sin^3 x$ satisfies $f(x) = \\frac{3}{4} \\sin x - \\frac{1}{4} \\sin 3x$, by the identity $\\sin 3x = 3 \\sin x - 4 \\sin^3 x$.\nSo its Fourier decomposition has only the frequencies $\\pm 1$ and $\\pm 3$.\nNon-example: writing a function as a power series $\\sum a_n x^n$ is an expansion into monomials, not into exponentials, so it is not Fourier analysis.\nThe word analysis refers to the decomposition step, and the word synthesis to the reconstruction step.\n",
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