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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[Hilbert spaces|HM2HHE3J51]] and let $K \\colon X \\to Y$ be a [[compact linear operator|M2NEP1K5AE]] with [[adjoint|6HSJBJ8TTQ]] $K^*$.\nLet $(\\sigma_n, v_n, u_n)_{n \\in N}$ be a singular system of $K$, with $N$ finite or $N = \\set{1, 2, \\dots}$.\nThis means that the $\\sigma_n \u003e 0$ are the nonzero [[singular values|QSJKPKPM7H]] of $K$, and $K v_n = \\sigma_n u_n$ and $K^* u_n = \\sigma_n v_n$.\nIt also means that $(v_n)$ is an orthonormal basis of $\\ker(K)^\\perp$ and $(u_n)$ is an orthonormal basis of the closure of $K(X)$.\nLet $y \\in Y$.\nThen the equation $K x = y$ has a solution $x \\in X$ if and only if\n\n1. $y \\in \\ker(K^*)^\\perp$, and\n2. $\\displaystyle \\sum_{n \\in N} \\frac{\\abs{\\ip{y}{u_n}}^2}{\\sigma_n^2} \u003c \\infty$.\n\nIn this case $x^\\dagger = \\sum_{n \\in N} \\sigma_n^{-1} \\ip{y}{u_n} v_n$ is a solution, and it is the unique solution of smallest norm.\n\n## Remarks\n\nProof idea: if $K x = y$, then $y \\perp \\ker(K^*)$ and $\\ip{y}{u_n} = \\ip{x}{K^* u_n} = \\sigma_n \\ip{x}{v_n}$, so the sum in clause 2 equals $\\sum_n \\abs{\\ip{x}{v_n}}^2 \\le \\norm{x}^2$ by Bessel's inequality.\nConversely, clause 2 makes the series for $x^\\dagger$ converge.\nThen $K x^\\dagger = \\sum_n \\ip{y}{u_n} u_n$ is the orthogonal projection of $y$ onto the closure of $K(X)$, which is $\\ker(K^*)^\\perp$.\nClause 1 makes it equal to $y$.\nThe solution $x^\\dagger$ lies in $\\ker(K)^\\perp$, so it has the smallest norm among all solutions.\n\nExample: the Volterra operator $(V x)(t) = \\int_0^t x(s) \\, ds$ on $L^2(0, 1)$ is injective with dense range.\nIts singular system is $\\sigma_n = \\frac{2}{(2n - 1)\\pi}$, $v_n(s) = \\sqrt{2} \\cos\\big((n - \\tfrac{1}{2}) \\pi s\\big)$ and $u_n(t) = \\sqrt{2} \\sin\\big((n - \\tfrac{1}{2}) \\pi t\\big)$.\nSo $V x = y$ is solvable if and only if $\\sum_n (2n - 1)^2 \\abs{\\ip{y}{u_n}}^2 \u003c \\infty$, a condition on the decay of the coefficients of $y$ that expresses the differentiability of $y$.\n\n## Explanations\n\n### L1 human\n\nA compact operator shrinks the component of $x$ along the $n$-th singular direction by the factor $\\sigma_n$, and these factors tend to zero.\nTo undo it one must divide the components of the data by $\\sigma_n$.\nPicard's criterion says that the equation can be solved exactly when the result of these divisions still has finite energy.\nData whose components do not decay fast enough, for example noisy data, are not in the range.\n\n### L2 human\n\nThe criterion is the infinite-dimensional form of solving a linear system by the singular value decomposition, see [[singular value decomposition|000NPTMX6J]].\nIt shows the [[ill-posedness of first-kind equations|KHXYWTNN2R]] in a quantitative way: an error of size $\\delta$ in the coefficient $\\ip{y}{u_n}$ becomes an error $\\delta / \\sigma_n$ in the solution.\nThe faster the singular values decay, the smoother the data must be, and the more ill posed the problem.\nRegularization methods such as truncated singular value expansion and Tikhonov regularization replace the factors $1/\\sigma_n$ by bounded filter factors.\nThe map $y \\mapsto x^\\dagger$ is the [[Moore–Penrose inverse|0YY8VX28JR]] of $K$ on its natural domain.\n",
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