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  "body": "\n## Statement\n\nLet $d \\ge 1$ and let $S \\subset \\R^d$ be a compact set of positive measure whose boundary has Lebesgue measure zero.\nLet $PW_S$ be the space of [[band-limited functions|P3T104G564]] with spectrum in $S$: the $f \\in L^2(\\R^d)$ whose [[Fourier transform|DPVM34ADZV]] $\\hat f(\\xi) = \\int f(x) e^{-i x \\cdot \\xi} \\, dx$ vanishes outside $S$.\nLet $\\Lambda \\subset \\R^d$ be uniformly discrete, that is, $\\inf \\set{\\abs{\\lambda - \\lambda'} : \\lambda \\ne \\lambda' \\in \\Lambda} \u003e 0$.\nDefine the lower and upper Beurling densities\n$$D^-(\\Lambda) = \\liminf_{r \\to \\infty} \\inf_{x \\in \\R^d} \\frac{\\#(\\Lambda \\cap (x + [0, r]^d))}{r^d}, \\qquad D^+(\\Lambda) = \\limsup_{r \\to \\infty} \\sup_{x \\in \\R^d} \\frac{\\#(\\Lambda \\cap (x + [0, r]^d))}{r^d}.$$\n\n1. If $\\Lambda$ is a sampling set for $PW_S$, that is, there are $0 \u003c A \\le B$ with $A \\norm{f}_2^2 \\le \\sum_{\\lambda \\in \\Lambda} \\abs{f(\\lambda)}^2 \\le B \\norm{f}_2^2$ for all $f \\in PW_S$, then $D^-(\\Lambda) \\ge \\abs{S} / (2\\pi)^d$.\n2. If $\\Lambda$ is an interpolation set for $PW_S$, that is, for every square-summable $(a_\\lambda)_{\\lambda \\in \\Lambda}$ there is $f \\in PW_S$ with $f(\\lambda) = a_\\lambda$ for all $\\lambda$, then $D^+(\\Lambda) \\le \\abs{S} / (2\\pi)^d$.\n\nHere $\\abs{S}$ is the Lebesgue measure of $S$.\n\n## Remarks\n\nHenry Landau proved these conditions in 1967.\nThe proof compares the eigenvalues of two concentration operators: the operator that restricts $f \\in PW_S$ to a large cube, and its discrete analogue on the samples in that cube.\nThe number of eigenvalues of the first operator that are not close to $0$ or $1$ is small compared with the volume of the cube.\nThe main count is $\\abs{S} r^d / (2\\pi)^d$.\n\nExample: for $d = 1$ and $S = [-\\pi, \\pi]$, the integers form a sampling and interpolation set, and $D^-(\\Z) = D^+(\\Z) = 1 = \\abs{S}/2\\pi$.\nThis is the critical case of the [[Nyquist–Shannon sampling theorem|BMXDMHRXBG]].\nSo uniform sampling at the Nyquist rate is optimal, and no irregular scheme can sample stably at a lower average rate.\n\nSince $f(\\lambda) = (2\\pi)^{-d} \\int_S \\hat f(\\xi) e^{i\\lambda \\cdot \\xi} \\, d\\xi$, the set $\\Lambda$ is a sampling set if and only if the exponentials $e^{i\\lambda \\cdot \\xi}$, $\\lambda \\in \\Lambda$, form a frame of $L^2(S)$.\nIt is a sampling and interpolation set if and only if they form a [[Riesz basis|ZYDYD7FB1Z]] of $L^2(S)$, and then both parts give $D^-(\\Lambda) = D^+(\\Lambda) = \\abs{S}/(2\\pi)^d$.\n\n## Explanations\n\n### L1 human\n\nTo record a signal whose frequencies lie in a set $S$, one measures its values at a set of points.\nLandau's theorem says that stable recording needs, on average, at least $\\abs{S}/(2\\pi)^d$ points per unit volume, however cleverly the points are placed.\nAnd one cannot prescribe arbitrary values at more than that many points per unit volume.\n\n### L2 human\n\nThe Shannon sampling theorem gives a sufficient condition for one regular lattice.\nLandau's theorem gives a necessary condition for every uniformly discrete set, measured by the Beurling densities.\nIt shows that the Nyquist rate is sharp also for irregular sampling.\nIt also implies that a Riesz basis of exponentials for $L^2(S)$ must have uniform density exactly $\\abs{S}/(2\\pi)^d$, which links it with the theory of [[nonharmonic Fourier series|TKNA7PR99K]].\nThe theorem has analogues for Gabor systems and for other spaces of functions with localized transforms.\n",
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