## Abstract

We consider the problem of finding a best uniform approximation to the standard monomial on the unit ball in ℂ^{2} by polynomials of lower degree with complex coefficients. We reduce the problem to a one-dimensional weighted minimization problem on an interval. In a sense, the corresponding extremal polynomials are uniform counterparts of the classical orthogonal Jacobi polynomials. They can be represented by means of special conformal mappings on the so-called comb-like domains. In these terms, the value of the minimal deviation and the representation for a polynomial of best approximation for the original problem are given. Furthermore, we derive asymptotics for the minimal deviation.

Original language | English |
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Pages (from-to) | 13-24 |

Number of pages | 12 |

Journal | Computational Methods and Function Theory |

Volume | 11 |

Issue number | 1 |

DOIs | |

State | Published - 2011 |

Externally published | Yes |

## Keywords

- Asymptotics
- Conformal mapping
- Minimal deviation
- Polynomial approximation
- Several variables
- Uniform norm

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