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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 [[#References|[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|Chebyshev system]]. For such systems the [[Chebyshev theorem|Chebyshev theorem]] and the [[De la Vallée-Poussin theorem|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Haar,   "Die Minkowskische Geometrie and die Annäherung an stetige Funktionen"  ''Math. Ann.'' , '''78'''  (1918)  pp. 249–311</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,   "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,   "Introduction to approximation theory" , McGraw-Hill  (1966)  pp. Chapt. 3</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.S.B. Holland,   B.N. Sahney,   "The general problem of approximation and spline functions" , R.E. Krieger  (1979)  pp. Chapt. 2</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.G. Lorentz,   S.D. Riemenschneider,   "Approximation and interpolation in the last 20 years" , ''Birkhoff interpolation'' , Addison-Wesley  (1983)  pp. xix-lv; in particular, xx-xxiii</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.F. Timan,   "Theory of approximation of functions of a real variable" , Pergamon  (1963)  pp. Chapt. 2  (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.R. Rice,   "The approximation of functions" , '''1. Linear theory''' , Addison-Wesley  (1964)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Meinardus,   "Approximation of functions: theory and numerical methods" , Springer  (1967)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D.S. Bridges,   "Recent developments in constructive approximation theory"  A.S. Troelstra (ed.)  D. van Dalen (ed.) , ''The L.E.J. Brouwer Centenary Symposium'' , ''Studies in logic'' , '''110''' , North-Holland  (1982)  pp. 41–50</TD></TR></table>