Optimal subspaces for mean square approximation of classes of differentiable functions on a segment

In this paper, we specify a set of optimal subspaces for $L_2$ approximation of three classes of functions in the Sobolev spaces $W_2^{(r)}$, defined on a segment and subject to certain boundary conditions. A subspace $X$ of dimension not exceeding n is called optimal for a function class $A$ if the best approximation of $A$ by $X$ equals the Kolmogorov $n$-width of $A$. These boundary conditions correspond to subspaces of periodically extended functions with symmetry properties. All of the approximating subspaces are generated by equidistant shifts of a single function. The conditions of optimality are given in terms of Fourier coefficients of a generating function. In particular, we indicate optimal spline spaces of all degrees $d \geqslant r-1$ with equidistant knots of several different types.