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 <!-- MATH ${\mathbf{N}}_0^{p,n}$ --> (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 <!-- MATH ${\mathbf{N}}_0^{p,n}$ --> we define a derivative <!-- MATH ${D^{n,\sigma }}f$ --> in <!-- MATH ${L^\sigma },\sigma = {(n + {p^{ - 1}})^{ - 1}}$ --> , which agrees with for smooth f, and prove several properties of <!-- MATH ${D^{n,\sigma }}$ --> .