Exact asymptotics of the optimal Lp-error of asymmetric linear spline approximation

In this paper we study the best asymmetric (sometimes also called penalized or sign-sensitive) approximation in the metrics of the space Lp, 1 6 p 6 ∞, of functions f ∈ C 2 [0,1] 2 � with nonnegative Hessian by piecewise linear splines s ∈ S(△N), generated by given triangulations △N with N elements. We find the exact asymptotic behavior of the optimal (over triangulations △N and splines s ∈ S(△N)) error of such an approximation as N → ∞.