Iterated kernel

Remarks

Example: $K(x, s) = x s$ on $[0, 1]$. Then $K_2(x, s) = \int_0^1 x t^2 s \, dt = x s / 3$, and by induction $K_n(x, s) = x s / 3^{n-1}$.

Non-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.

Each $K_n$ is continuous, and $K_n$ is the kernel of the $n$-th power of the integral operator of $K$. More generally $K_{n + m}(x, s) = \int_a^b K_n(x, t) K_m(t, s) \, dt$ for $n, m \ge 1$. If $\abs{K} \le M$, then $\abs{K_n(x, s)} \le M^n (b - a)^{n - 1}$, by induction on $n$. The same definition is used for square-integrable kernels, with equality almost everywhere. For a Volterra kernel, which vanishes for $s > x$, the integral runs over $s \le t \le x$.