Sharp Estimates for Mean Square Approximations of Classes of Differentiable Periodic Functions by Shift Spaces

Let L₂ be the space of 2π-periodic square-summable functions and E(f, X)₂ be the best approximation of f by the space X in L₂. For n ∈ ? and B ∈ L₂, let ({{Bbb S}_{B,n}}) be the space of functions s of the form (sleft( x right) = sumlimits_{j = 0}^{2n - 1} {{beta _j}Bleft( {x - frac{{jpi }}{n}} right)} ). This paper describes all spaces ({{Bbb S}_{B,n}}) that satisfy the exact inequality (E{left( {f,{S_{B,n}}} right)_2} leqslant frac{1}{{^{{n^r}}}}parallel {f^{left( r right)}}{parallel _2}). (2n–1)-dimensional subspaces fulfilling the same estimate are specified. Well-known inequalities are for approximation by trigonometric polynomials and splines obtained as special cases.