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 ∈ ℕ and B ∈ L₂, let S_{B , n} be the space of functions s of the form s(x)=∑j=02n−1βjB(x−jπn). This paper describes all spaces S_{B , n} that satisfy the exact inequality E(f,SB,n)2≤1nr∥f(r)∥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.