  Cauchy Biorthogonal Polynomials appear in the study of special solutions to
the dispersive nonlinear partial differential equation called the
Degasperis-Procesi (DP) equation, as well as in certain two-matrix random
matrix models. Another context in which such biorthogonal polynomials play a
role is the cubic string; a third order ODE boundary value problem -f'''=zg f
which is a generalization of the inhomogeneous string problem studied by M.G.
Krein. A general class of such boundary value problems going beyond the
original cubic string problem associated with the DP equation is discussed
under the assumption that the source of inhomogeneity g is a discrete measure.
It is shown that by a suitable choice of a generalized Fourier transform
associated to these boundary value problems one can establish a Parseval type
identity which aligns Cauchy biorthogonal polynomials with certain natural
orthogonal systems on L^2_g.
