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  "body": "\n## Definition\n\nLet $a \u003c b$ and let $K \\colon [a, b] \\times [a, b] \\to \\C$ be a [[continuous kernel|ZM5G144PZG]].\nThe *iterated kernels* of $K$ are the functions $K_n \\colon [a, b] \\times [a, b] \\to \\C$, for $n \\ge 1$, defined by $K_1 = K$ and\n$$K_{n}(x, s) = \\int_a^b K(x, t) \\, K_{n-1}(t, s) \\, dt \\quad \\text{for } n \\ge 2.$$\nThe function $K_n$ is the $n$-th iterated kernel of $K$.\n\n## Remarks\n\nExample: $K(x, s) = x s$ on $[0, 1]$.\nThen $K_2(x, s) = \\int_0^1 x t^2 s \\, dt = x s / 3$, and by induction $K_n(x, s) = x s / 3^{n-1}$.\n\nNon-example: the pointwise power $K(x, s)^2 = x^2 s^2$ is not the second iterated kernel $x s / 3$; iteration composes the integral operators, it does not multiply kernel values.\n\nEach $K_n$ is continuous, and $K_n$ is the kernel of the $n$-th power of the integral operator of $K$.\nMore generally $K_{n + m}(x, s) = \\int_a^b K_n(x, t) K_m(t, s) \\, dt$ for $n, m \\ge 1$.\nIf $\\abs{K} \\le M$, then $\\abs{K_n(x, s)} \\le M^n (b - a)^{n - 1}$, by induction on $n$.\nThe same definition is used for square-integrable kernels, with equality almost everywhere.\nFor a Volterra kernel, which vanishes for $s \u003e x$, the integral runs over $s \\le t \\le x$.\n",
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