On the Degree of Convergence of Piecewise Polynomial Approximation on Optimal Meshes

The degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain F-spaces N₀p,n (introduced for this purpose in [2]). We obtain two o-results and use pairs of inequalities of Bernstein- and Jackson-type to prove several direct and converse theorems. For f in N₀p,n we define a derivative Dn,σf in L^σ ,σ = (n + p- 1)- 1, which agrees with Dⁿf for smooth f, and prove several properties of Dn,σ.