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  "body": "\n## Definition\n\nLet $h_0 \\in \\R$ and, for each $n \\ge 1$, let $h_n \\colon [0, \\infty)^n \\to \\R$ be an integrable function, the *$n$-th Volterra kernel*.\nAssume that $\\sum_{n \\ge 1} \\norm{h_n}_{L^1} r^n \u003c \\infty$ for some $r \u003e 0$.\nFor a bounded measurable input $x \\colon \\R \\to \\R$ with $\\sup \\abs{x} \\le r$, the *Volterra series* with these kernels gives the output\n$$y(t) = h_0 + \\sum_{n=1}^{\\infty} \\int_0^\\infty \\!\\!\\cdots\\! \\int_0^\\infty h_n(\\tau_1, \\ldots, \\tau_n)\\, x(t - \\tau_1) \\cdots x(t - \\tau_n)\\, d\\tau_1 \\cdots d\\tau_n, \\quad t \\in \\R.$$\nThe $n$-th term is bounded by $\\norm{h_n}_{L^1}\\, r^n$, so the series converges absolutely and uniformly in $t$.\n\n## Remarks\n\nThe operator $x \\mapsto y$ is causal, since $y(t)$ depends only on $x(s)$ for $s \\le t$, and time-invariant, since shifting $x$ in time shifts $y$ by the same amount.\nThe kernels can be taken symmetric in their variables without changing $y$.\nIf only $h_1$ is nonzero, $y = h_0 + h_1 * x$ is a [[convolution|2Z29JN3VES]], the output of a linear system with impulse response $h_1$.\nSome authors also allow kernels with delta functions, or integrals over all of $\\R^n$ for noncausal systems.\n\nExample: let $z(t) = \\int_0^\\infty e^{-\\tau} x(t - \\tau)\\, d\\tau$ and $y = z^2$.\nThen $y(t) = \\int_0^\\infty \\int_0^\\infty e^{-\\tau_1 - \\tau_2}\\, x(t - \\tau_1)\\, x(t - \\tau_2)\\, d\\tau_1\\, d\\tau_2$ by Fubini's theorem.\nSo $y$ is a Volterra series with $h_2(\\tau_1, \\tau_2) = e^{-\\tau_1 - \\tau_2}$, of $L^1$ norm $1$, and all other kernels $0$.\n\nNon-example: the system $y(t) = t\\, x(t)$ is not given by any Volterra series, because it is not time-invariant.\nThe input $x = 1$ gives $y(t) = t$, which is not constant, while a Volterra series maps a constant input to a constant output.\n",
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