## Abstract

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

Original language | English |
---|---|

Pages (from-to) | 73-91 |

Number of pages | 19 |

Journal | Constructive Approximation |

Volume | 20 |

Issue number | 1 |

DOIs | |

State | Published - 2004 |

## Keywords

- Bivariate piecewise polynomials
- Bivariate rational functions
- Hardy-Littlewood maximal function
- Multivariate nonlinear approximation

## Fingerprint

Dive into the research topics of 'On the Relation between Piecewise Polynomial and Rational Approximation in L_{p}(R

^{2})'. Together they form a unique fingerprint.