Rayleigh quotient for symmetric kernels

Summary

Let $K$ be a Hermitian kernel with integral operator $T_K$. The Rayleigh quotient of $K$ at a nonzero function $\varphi$ is the real number $\ip{T_K \varphi}{\varphi} / \ip{\varphi}{\varphi}$. It is the integral analogue of the Rayleigh quotient of a Hermitian matrix. Its extreme values and critical points give the eigenvalues and eigenfunctions of $T_K$, which turns eigenvalue problems for integral equations into problems of optimization.