Difference kernel

Remarks

Example: $K(x, s) = \cos(x - s)$ with $k = \cos$.

Non-example: $K(x, s) = x s$ on $[0, 1]$. If $x s = k(x - s)$, then putting $s = x$ gives $x^2 = k(0)$ for all $x \in [0, 1]$, which is impossible.

On $J = \R$, the integral operator is the convolution $\int_{\R} k(x - s) \varphi(s) \, ds = (k * \varphi)(x)$. On $J = [0, \infty)$ with upper limit $x$, it is the one-sided convolution $\int_0^x k(x - s) \varphi(s) \, ds$, which the Laplace transform turns into a product. Equations with a difference kernel are called convolution equations. The equation $\varphi(x) = f(x) + \int_0^{\infty} k(x - s) \varphi(s) \, ds$ for $x \ge 0$ is the Wiener–Hopf equation.