Comments on eigenfunction expansions of discontinuous Sturm-Liouville systems (Q752935)

Consider the eigenvalue problem \(-y''+q(x)y=\lambda y\) on \(0\leq x\leq \pi\) satisfying the symmetric boundary conditions \(hy(0)- y'(0)=hy(\pi)+y'(\pi)=0\) with two symmetric discontinuities at \(x=d_ 1\) and \(x=d_ 2=\pi -d_ 1\), satisfying the jump conditions: \(y(d_ 1+)=ay(d_ 1-),\quad y(d_ 2-)=ay(d_ 2+),\quad y(d_ 1+)=a^{- 1}y(d_ 1-)+by(d_ 1-),\quad y'(d_ 2-)=a^{-1}y(d_ 2+)-by(d_ 2+),\) where \(a>0\) and b are the jumb constants, h is a boundary constant and q is integrable on [0,\(\pi\) ]. The authors shows that any function f such that f and \(f'\) are sectionally continuous on (0,\(\pi\)) with sections (0,d), (d,\(\pi\)-d), (\(\pi\)-d,\(\pi\)) has a Fourier series-type expansion in terms of the eigenfunctions of the above problem.