# Haar condition

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

A condition on continuous functions , , that are linearly independent on a bounded closed set of a Euclidean space. The Haar condition, stated by A. Haar , ensures for any continuous function on the uniqueness of the polynomial of best approximation in the system , that is, of the polynomial (*)

for which  The Haar condition says that any non-trivial polynomial of the form (*) can have at most distinct zeros on . For any continuous function on there exists a unique polynomial of best approximation in the system if and only if the system satisfies the Haar condition. A system of functions satisfying the Haar condition is called a Chebyshev system. For such systems the Chebyshev theorem and the de la Vallée-Poussin theorem (on alternation) hold. The Haar condition is sufficient for the uniqueness of the polynomial of best approximation in the system with respect to the metric of ( ) for any continuous function on .

How to Cite This Entry:
Haar condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Haar_condition&oldid=11643
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article