On Kellog's theorem for discontinuous Green functions (Q1326024)

The authors consider the integral operator, \(K_ q x(t) = \int^ b_ a K(t,s) q(s) x(s)ds\), \(q>0\), for a class of piecewise continuous kernels \(K(t,s)\). They prove that the results of \textit{O. D. Kellog} [The oscillation of functions of an orthogonal set. Am. J. Math. 38, 1-5 (1916; JFM 46.0647.01) and Orthogonal function sets arising from integral equations. 40, 145-154 (1918; JFM 46.0648.01)], concerning the oscillating properties of the eigenfunctions when the kernel \(K(t,s)\) is continuous, remain true.