BATEMANS EQUATION AND SIMILAR UNITS

The multiplication K(x, y) circle F(y, z) = integral 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 multipl ication has a unit element, i.e., a kernel E(x, y) such that E(x, y) c ircle F(y, z) = F(x, z) or integral E(r, y)f(y) dy = f(x) for infinite ly many linear independent functions f. Bateman's function [sin(x - y) ]/pi(x - y) is an example of such a kernel E(x, y). This paper develop s a procedure to construct Bateman's function and similar units. (C) E lsevier Science Inc., 1997