Bateman's equation and similar units

The multiplication K(x, y)° ○ F(y, z) = ∫K(x, y)F(y, z) dy of real functions K and F can be interpreted as the analytic version of matrix multiplication. This suggests examining whether this multiplication has a unit element, i.e., a kernel E(x, y) such that E(x, y)○ F(y,z} = F(x, z) or ∫E(x, y)f(y) dy = f(x) for infinitely many linear independent functions f. Bateman's function [sin(x - y)]/π(x - y) is an example of such a kernel E(x, y). This paper develops a procedure to construct Bateman's function and similar units.