Sharp estimates for mean square approximations of classes of differentiable periodic functions by shift spaces

Let L2 be the space of 2 -periodic square integrable functions; E(f,X)2 is the best approximation of the function f by the space X in L2. For n ∈ N, B ∈ L2 we denote by SB,n the space of functions s of the form s(x) = 2n−1 Xj=0 jB x − j n . In this paper, we give a description of all spaces SB,n for which the sharp inequality E(f, SB,n)2 6 1 nr kf(r)k2 holds. In doing so, we indicate the subspaces of dimension 2n − 1 that provide the same estimate. The well-known inequalities for approximation by trigonometric polynomials and splines are obtained as particular cases