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A condition on continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h0460501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h0460502.png" />, that are linearly independent on a bounded closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h0460503.png" /> of a Euclidean space. The Haar condition, stated by A. Haar [[#References|[1]]], ensures for any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h0460504.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h0460505.png" /> the uniqueness of the polynomial of best approximation in the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h0460506.png" />, that is, of the polynomial
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h0460507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$P_{n-1}(t)=\sum_{k=1}^nc_kx_k(t)\tag{*}$$
  
 
for which
 
for which
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h0460508.png" /></td> </tr></table>
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$$\max_{t\in M}|f(t)-P_{n-1}(t)|=$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h0460509.png" /></td> </tr></table>
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$$=\min_{\{a_k\}}\max_{t\in M}\left|f(t)-\sum_{k=1}^na_kx_k(t)\right|.$$
  
The Haar condition says that any non-trivial polynomial of the form (*) can have at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h04605010.png" /> distinct zeros on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h04605011.png" />. For any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h04605012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h04605013.png" /> there exists a unique polynomial of best approximation in the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h04605014.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h04605015.png" /> with respect to the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h04605016.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h04605017.png" />) for any continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046050/h04605018.png" />.
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The Haar condition says that any non-trivial polynomial of the form \ref{*} 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====
 
====References====

Revision as of 17:57, 28 June 2015

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

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

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 \ref{*} 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)


Comments

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
How to Cite This Entry:
Haar condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Haar_condition&oldid=36524
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article