On the Relation between Piecewise Polynomialand Rational Approximation in Lp(R2)

In the univariate case there are certain equivalences between thenonlinear approximation methods that use piecewise polynomials andthose that use rational functions. It is known that for certainparameters the respective approximation spaces are identical andmay be described as Besov spaces. The characterization of theapproximation spaces of the multivariate nonlinear approximationby piecewise polynomials and by rational functions is not known.In this work we compare between the two methods in the bivariatecase. We show some relations between the approximation spaces ofpiecewise polynomials defined on n triangles and those ofbivariate rational functions of total degree n which aredescribed by n parameters. Thus we compare two classes ofapproximants with the same number Cn of parameters. We considerthis the proper comparison between the two methods.