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)

References

[a1]	E.W. Cheney, "Introduction to approximation theory" , McGraw-Hill (1966) pp. Chapt. 3
[a2]	A.S.B. Holland, B.N. Sahney, "The general problem of approximation and spline functions" , R.E. Krieger (1979) pp. Chapt. 2
[a3]	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
[a4]	A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) pp. Chapt. 2 (Translated from Russian)
[a5]	J.R. Rice, "The approximation of functions" , 1. Linear theory , Addison-Wesley (1964)
[a6]	G. Meinardus, "Approximation of functions: theory and numerical methods" , Springer (1967)
[a7]	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