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  "body": "\n## Statement\n\nLet $B \u003e 0$.\nLet $f, g \\in L^2(\\R)$ be continuous [[band-limited functions|P3T104G564]] with bandwidth $B$, that is, their [[Fourier transforms|DPVM34ADZV]] vanish almost everywhere outside $[-B, B]$.\nIf $f(k / (2B)) = g(k / (2B))$ for every $k \\in \\Z$, then $f = g$.\n\n## Remarks\n\nThe difference $h = f - g$ is band-limited and vanishes at all points $k/(2B)$.\nOn $[-B, B]$, $\\hat h$ has the Fourier coefficients $\\frac{1}{2B} \\int_{-B}^{B} \\hat h(\\xi) e^{-2\\pi i k \\xi / (2B)}\\, d\\xi = \\frac{1}{2B} h(-\\frac{k}{2B})$ by Fourier inversion, and they are all zero.\nSo $\\hat h = 0$ in [[$L^2$|SASP14MA1Z]], and $h = 0$.\nThe [[Whittaker–Shannon interpolation formula|Z4ZFP6FNK2]] gives $f$ explicitly from its samples.\n\nThe rate $2B$ samples per unit length is called the Nyquist rate.\nIt cannot be lowered: if the spacing is $h \u003e 1/(2B)$, then $f(x) = \\sin(\\pi x / h)\\, \\operatorname{sinc}(2\\delta x)^2$ with $\\delta = \\frac12 (B - \\frac{1}{2h}) \u003e 0$ is a nonzero function, band-limited with bandwidth $B$, in $L^2(\\R)$, that vanishes at every point $kh$.\nHere $\\operatorname{sinc}(t) = \\sin(\\pi t)/(\\pi t)$, and $\\operatorname{sinc}(2\\delta x)^2$ has transform supported in $[-2\\delta, 2\\delta]$.\n\nThe theorem is named after H. Nyquist and C. E. Shannon; E. T. Whittaker and V. A. Kotelnikov found equivalent results.\n\n## Explanations\n\n### L1 human\n\nIf a signal has no frequencies above $B$ cycles per second, then sampling it $2B$ times per second loses nothing: the samples determine the signal.\nSampling more slowly can confuse a high frequency with a lower one, which is called aliasing.\nThis is why audio is sampled at a little more than twice the highest audible frequency.\n\n### L2 human\n\nThe proof is a duality between the line and the circle.\nThe samples of $f$ are the Fourier coefficients of $\\hat f$, viewed as a periodic function on an interval of length $2B$.\nEquivalently, multiplying $f$ by a Dirac comb of spacing $1/(2B)$ periodizes $\\hat f$ with period $2B$, and the copies do not overlap exactly when the band fits.\nLandau's density theorem extends the necessity of the Nyquist rate to irregular sampling sets.\n",
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