Optimal subspaces for mean square approximation of classes of differentiable functions on the half-line

We obtain sharp inequalities for the best mean square approximation of two classes of functions on the half-line, defined by boundary conditions corresponding to even and odd extension of a function. Optimal subspaces are provided by even and odd parts of the spaces generated by equidistant shifts of a single function. Under certain additional conditions on this function, the sharpness of the inequalities in the sense of average widths is proved.