On the Exact Asymptotics of the Best Relative Approximations of Classes of Periodic Functions by Splines

Abstract

We obtain the exact asymptotics (as n → ∞) of the best L ₁-approximations of classes \(W_1^r\) of periodic functions by splines s ∈ S _{2n, r − 1} and s ∈ S _{2n, r + k − 1} (S _{2n, r} is the set of 2π-periodic polynomial splines of order r and defect 1 with nodes at the points kπ/n, k ∈ Z) under certain restrictions on their derivatives.