# Polynomial of best approximation

A polynomial furnishing the best approximation of a function in some metric, relative to all polynomials constructed from a given (finite) system of functions. If is a normed linear function space (such as or , ), and if

is a system of linearly independent functions in , then for any the (generalized) polynomial of best approximation

(*) |

defined by the relation

exists. The polynomial of best approximation is unique for all if is a space with a strictly convex norm (i.e. if and , then ). This is the case for , . In , which has a norm that is not strictly convex, the polynomial of best approximation for any is unique if is a Chebyshev system on , i.e. if each polynomial

has at most zeros on . In particular, one has uniqueness in the case of the (usual) algebraic polynomials in , and also for the trigonometric polynomials in the space of continuous -periodic functions on the real line, with the uniform metric. If the polynomial of best approximation exists and is unique for any , it is a continuous function of .

Necessary and sufficient conditions for a polynomial to be a best approximation in the spaces and are known. For example, one has Chebyshev's theorem: If is a Chebyshev system, then the polynomial (*) is a polynomial of best approximation for a function in the metric of if and only if there exists a system of points , , at which the difference

assumes values

and, moreover,

The polynomial (*) is a polynomial of best approximation for a function , , in the metric of that space, if and only if for ,

In , the conditions

are sufficient for to be a polynomial of best approximation for , and if the measure of the set of all points at which is zero, they are also necessary; see also Markov criterion.

There exist algorithms for the approximate construction of polynomials of best uniform approximation (see e.g. [3], [5]).

#### References

[1] | N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian) |

[2] | N.P. Korneichuk, "Extremal problems in approximation theory" , Moscow (1976) (In Russian) |

[3] | V.K. Dzyadyk, "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow (1977) (In Russian) |

[4] | V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian) |

[5] | P.J. Laurent, "Approximation et optimisation" , Hermann (1972) |

[6] | E.Ya. Remez, "Foundations of numerical methods of Chebyshev approximation" , Kiev (1969) (In Russian) |

#### Comments

#### References

[a1] | A.M. Pinkus, "On -approximation" , Cambridge Univ. Press (1989) |

[a2] | A. Schönhage, "Approximationstheorie" , de Gruyter (1971) |

[a3] | G.A. Watson, "Approximation theory and numerical methods" , Wiley (1980) |

[a4] | E.W. Cheney, "Introduction to approximation theory" , McGraw-Hill (1966) pp. Chapts. 4&6 |

[a5] | M.J.D. Powell, "Approximation theory and methods" , Cambridge Univ. Press (1981) |

[a6] | J.R. Rice, "The approximation of functions" , 1. Linear theory , Addison-Wesley (1964) |

[a7] | T.J. Rivlin, "An introduction to the approximation of functions" , Blaisdell (1969) |

**How to Cite This Entry:**

Polynomial of best approximation.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Polynomial_of_best_approximation&oldid=18661