Optimal subspaces for mean square approximation of classes of differentiable functions with boundary conditions

In this paper, we specify a set of optimal subspaces for $L_2$ approximation of three classes of functions in the Sobolev space $W_2^{(r)}$, defined on a segment and subject to certain boundary conditions. All of these subspaces are generated by equidistant shifts of a single function. In particular, we indicate optimal spline spaces of all degrees $d⩾r-1$ with uniform knots.