The Fox–Li operator is a convolution operator over a finite
interval with a special highly oscillatory kernel. It plays an important
role in laser engineering. However, the mathematical analysis of its spectrum
is still rather incomplete. In this expository paper we survey part
of the state of the art, and our emphasis is on showing how standard
Wiener–Hopf theory can be used to obtain insight into the behaviour of
the singular values of the Fox–Li operator. In addition, several approximations
to the spectrum of the Fox–Li operator are discussed and results
on the singular values and eigenvalues of certain related operators are
derived
