Haar condition
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
[1],
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
 .
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) |
Yu.N. Subbotin
CommentsReferences| [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 |
This text originally appeared in Encyclopaedia of Mathematics
- ISBN 1402006098
|