Let σ > 0, m, r ∈ ℕ, m ≥ r, let S_σ,m be the space of splines of order m and minimal defect with nodes jπσ (j ∈ ℤ), and let A_σ,m(f)_p be the best approximation of a function f by the set S_σ,m in the space L_p(ℝ). It is known that for p = 1,+∞, (Formula presented.) where K_r are the Favard constants. In this paper, linear operators X_σ,r,m with values in S_σ,m such that for all p ∈ [1,+∞] and f ∈ W_p ^(r)(ℝ),ǁf−X_σ,r,m(f)ǁ_p≤K_r/σ_rǁf^(r)ǁp are constructed. This proves that the upper bounds indicated above can be achieved by linear methods of approximation, which was previously unknown. Bibliography: 21 titles.