Generalised Fourier methods for 1st-order distributed systems

Eigenfunction expansion of linear distributed systems is only possible when discrete solutions exist to the spatial-operator eigenvalue problem. 1st-order systems are naturally associated with a continuous spectrum, and a reduction to a finite lumped-system approximation is accordingly difficult. When the extended operator concept is used to introduce artificial periodicity, however, a discrete Fourier expansion becomes possible, yielding state-vector equations in the expansion coefficients. This generalised Fourier method particularly applies to distributed systems involving 1st-order spatial operators with 2-point boundary values.