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  "body": "\n## Definition\n\nLet $c \\ge 0$ be a real parameter, called the bandwidth parameter.\nConsider the differential equation on $(-1, 1)$\n$$\\frac{d}{dx}\\Big( (1 - x^2) \\frac{d\\psi}{dx} \\Big) - c^2 x^2 \\psi = -\\chi \\psi,$$\nwhere $\\chi$ is a real number.\nThere is an increasing sequence $\\chi_0 \u003c \\chi_1 \u003c \\chi_2 \u003c \\cdots$ that tends to $\\infty$ and has the following property.\nThe equation has a nonzero bounded solution on $(-1, 1)$ exactly when $\\chi = \\chi_m$ for some $m$.\nFor each $\\chi_m$ this solution is unique up to a constant factor.\nThe *prolate spheroidal wave functions* with parameter $c$ are these solutions $\\psi_0, \\psi_1, \\psi_2, \\dots$, normalized by $\\int_{-1}^1 \\abs{\\psi_m(x)}^2\\, dx = 1$ and taken real.\n\n## Remarks\n\nExample: for $c = 0$ the equation is Legendre's equation $((1 - x^2)\\psi')' = -\\chi \\psi$.\nIts bounded solutions are the Legendre polynomials $P_m$ with $\\chi_m = m(m + 1)$, so $\\psi_m = \\sqrt{(2m + 1)/2}\\, P_m$; for instance $\\psi_0 = 1/\\sqrt{2}$ satisfies the equation with $\\chi_0 = 0$.\n\nNon-example: for $c = 0$ and $\\chi = 0$, the function $\\psi(x) = \\frac12 \\log \\frac{1 + x}{1 - x}$ also solves the equation, since $(1 - x^2)\\psi'(x) = 1$.\nIt is not bounded near $\\pm 1$, so it is not a prolate spheroidal wave function.\n\nFor $c \u003e 0$, the functions $\\psi_m$ are also the [[eigenfunctions|0BQAA2BK3Q]] of the integral operator\n$$Q_c \\psi(x) = \\int_{-1}^1 \\frac{\\sin c(x - y)}{\\pi (x - y)}\\, \\psi(y)\\, dy$$\non $L^2(-1, 1)$, which is [[self-adjoint|0NEGN5CRSC]], positive and [[compact|M2NEP1K5AE]].\nIts eigenvalues satisfy $1 \u003e \\lambda_0 \u003e \\lambda_1 \u003e \\cdots \u003e 0$.\nThe kernel of $Q_c$ is that of the projection onto functions whose Fourier transform, in the convention $\\int f(t) e^{-i\\omega t}\\, dt$, vanishes outside $[-c, c]$.\nExtending $\\psi_m$ to $\\R$ by the same integral, divided by $\\lambda_m$, gives a [[band-limited function|P3T104G564]], and $\\lambda_m$ is the fraction of its energy that lies in $[-1, 1]$.\n",
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