  We consider the boundary value problems (BVPs) for linear secondorder ODEs
with a strongly positive operator coefficient in a Banach space. The solutions
are given in the form of the infinite series by means of the Cayley transform
of the operator, the Meixner type polynomials of the independent variable, the
operator Green function and the Fourier series representation for the
right-hand side of the equation. The approximate solution of each problem is a
partial sum of N (or expressed through N) summands. We prove the weighted error
estimates depending on the discretization parameter N, the distance of the
independent variable to the boundary points of the interval and some smoothness
properties of the input data.
