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  "body": "\n## Definition\n\nLet $a \u003c b$.\nA *degenerate kernel* on $[a, b]$ is a [[kernel|GSMBXT7FPK]] $K \\colon [a, b] \\times [a, b] \\to \\C$ for which there are an integer $n \\ge 1$ and functions $a_1, \\ldots, a_n, b_1, \\ldots, b_n \\colon [a, b] \\to \\C$ such that\n$$K(x, s) = \\sum_{i=1}^{n} a_i(x) \\, b_i(s) \\quad \\text{for all } x, s \\in [a, b].$$\n\n## Remarks\n\nExample: $K(x, s) = \\cos(x - s) = \\cos x \\cos s + \\sin x \\sin s$, with $n = 2$, $a_1 = b_1 = \\cos$ and $a_2 = b_2 = \\sin$.\n\nNon-example: $K(x, s) = e^{x s}$ on $[0, 1]$.\nIf $e^{x s} = \\sum_{i=1}^{n} a_i(x) b_i(s)$, then each function $x \\mapsto e^{x s}$ would lie in the span of $a_1, \\ldots, a_n$.\nBut the functions $e^{s_1 x}, \\ldots, e^{s_{n+1} x}$ for distinct $s_j$ are [[linearly independent|0CRMGPAH83]], which gives $n + 1$ independent functions in a space of dimension at most $n$.\n\nThe names *separable kernel* and *kernel of finite rank* are also used.\nOne usually asks that $a_i$ and $b_i$ be continuous, or square-integrable, and one may take $a_1, \\ldots, a_n$ and $b_1, \\ldots, b_n$ linearly independent.\nFor $\\varphi(x) = f(x) + \\lambda \\int_a^b K(x, s) \\varphi(s) \\, ds$ with a degenerate $K$, every solution has the form $\\varphi = f + \\lambda \\sum_i c_i a_i$ with $c_i = \\int_a^b b_i \\varphi \\, ds$, and the $c_i$ solve an $n \\times n$ linear system.\n",
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