Extension of Hilbert-Schmidt's theorem to a certain class of problems with unsymmetric kernels (Q799921)

Hilbert-Schmidt's theorem governs the expandibility of a given function, which is representable in terms of an integral involving a symmetric kernel together with a certain generating function, and this expansion is in terms of the kernel's eigenfunctions. In this paper it is shown, that for certain unsymmetric kernels, which have a bilinear expansion involving an appropriate set of eigenfunctions and their adjoints, a corresponding expansion theorem for a given function can be derived, if this function admits an integral representation by means of the unsymmetric kernel and again a certain generating function. Yet, the expandibility depends now on the magnitude of a perturbing parameter, which caused the unsymmetric kernel to orginate from a symmetric kernel. Since the range of magnitude of the perturbing parameter, in which expandibility is warranted, is the range of stability of a related mechanical problem, the new expansion theorem has practical relevance. An application of this new theorem for the derivation of ''fundamental integrals'' for certain mechanical problems having ''non-selfadjoint differential equations'' has been given in the meantime in Hadronic J. 6, 1488-1508 (1983).