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  "body": "\n## Statement\n\nLet $B \u003e 0$, and let $f \\in L^2(\\R)$ be a continuous [[band-limited function|P3T104G564]] with bandwidth $B$, that is, its [[Fourier transform|DPVM34ADZV]] vanishes almost everywhere outside $[-B, B]$.\nLet $\\operatorname{sinc}(t) = \\sin(\\pi t)/(\\pi t)$ for $t \\ne 0$ and $\\operatorname{sinc}(0) = 1$.\nThen\n$$f(x) = \\sum_{k \\in \\Z} f\\Big( \\frac{k}{2B} \\Big)\\, \\operatorname{sinc}(2Bx - k) \\quad \\text{for every } x \\in \\R,$$\nwhere the series converges in [[$L^2(\\R)$|SASP14MA1Z]] and uniformly on $\\R$.\n\n## Remarks\n\nThe proof expands $\\hat f$, restricted to $[-B, B]$, in a [[Fourier series|G12KXDHPPS]] of period $2B$.\nWriting $\\hat f(\\xi) = \\sum_k c_k e^{-2\\pi i k \\xi/(2B)}$ on $[-B, B]$, Fourier inversion gives $c_k = \\frac{1}{2B} f(\\frac{k}{2B})$.\nInverting term by term, the inverse transform of $e^{-2\\pi i k \\xi/(2B)} \\mathbf{1}_{[-B, B]}(\\xi)$ is $2B \\operatorname{sinc}(2Bx - k)$, which gives the formula.\nThe functions $\\sqrt{2B}\\, \\operatorname{sinc}(2Bx - k)$, $k \\in \\Z$, form an orthonormal basis of the band-limited functions of bandwidth $B$, so the series converges in $L^2$, with $\\sum_k \\abs{f(k/(2B))}^2 = 2B \\norm{f}_2^2$ by the [[Plancherel theorem|HPZHRZ0502]].\nUniform convergence follows from the bound $\\sup \\abs{g} \\le \\sqrt{2B} \\norm{g}_2$ for band-limited $g$ of bandwidth $B$.\n\nExample: for $f = \\operatorname{sinc}$, $B = \\frac12$, the samples are $f(k) = 0$ for $k \\ne 0$ and $f(0) = 1$, and the formula reduces to $f(x) = \\operatorname{sinc}(x)$.\n\nThe sample spacing $1/(2B)$ cannot be enlarged; if the band is wider than $[-B, B]$, different functions can have the same samples, which is aliasing.\n\n## Explanations\n\n### L1 human\n\nA signal with no frequencies above $B$ can be rebuilt exactly from its values at the times $k/(2B)$.\nEach sample is replaced by a sinc pulse centred at its time and scaled by its value, and the pulses are added.\nEach pulse is $1$ at its own sample time and $0$ at all the others, so the sum has the right values at the sample points.\nThe band limit forces it to be right everywhere in between.\n\n### L2 human\n\nThis is the explicit form of the sampling theorem.\nIt is also the expansion of $f$ in an orthonormal basis of the Paley–Wiener space, made of translates of one sinc function.\nIn practice the slow decay of sinc makes truncation errors large, so oversampling and smoother interpolation kernels are used.\n",
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