On the expansion problem of a function associated with a system of second order differential equations

In the present paper we generalize the Parseval theorem and the expansion theorem for a function f(x) = (f 1 (x), f 2 (x)) T satisfying f T (x)R(x)f(x) ∈ L (- ∞ , ∞ ) associated with the above system of second order differential equations under the general boundary conditions, where the elements of R(x) i.e. s(x), t(x) are assumed to be greater than zero for x ∈ (a, b), a, b being finite or infinite.