# Haar condition

A condition on continuous functions $x_k$, $k=1,\dots,n$, that are linearly independent on a bounded closed set $M$ of a Euclidean space. The Haar condition, stated by A. Haar [1], ensures for any continuous function $f$ on $M$ the uniqueness of the polynomial of best approximation in the system $\{x_k\}$, that is, of the polynomial

$$P_{n-1}(t)=\sum_{k=1}^nc_kx_k(t)\label{*}\tag{*}$$

for which

$$\max_{t\in M}|f(t)-P_{n-1}(t)|=$$

$$=\min_{\{a_k\}}\max_{t\in M}\left|f(t)-\sum_{k=1}^na_kx_k(t)\right|.$$

The Haar condition says that any non-trivial polynomial of the form \eqref{*} can have at most $n-1$ distinct zeros on $M$. For any continuous function $f$ on $M$ there exists a unique polynomial of best approximation in the system $\{x_k\}_{k=1}^n$ 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 $\{x_k\}_{k=1}^n$ with respect to the metric of $L[a,b]$ ($M=[a,b]$) for any continuous function on $[a,b]$.

#### References

 [1] A. Haar, "Die Minkowskische Geometrie and die Annäherung an stetige Funktionen" Math. Ann. , 78 (1918) pp. 249–311 [2] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)