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A polynomial furnishing the best approximation of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p0737301.png" /> in some metric, relative to all polynomials constructed from a given (finite) system of functions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p0737302.png" /> is a normed linear function space (such as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p0737303.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p0737304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p0737305.png" />), and if
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<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/p/p073/p073730/p0737306.png" /></td> </tr></table>
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is a system of linearly independent functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p0737307.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p0737308.png" /> the (generalized) polynomial of best approximation
+
A polynomial furnishing the best approximation of a function  $  x ( t) $
 +
in some metric, relative to all polynomials constructed from a given (finite) system of functions. If  $  X $
 +
is a normed linear function space (such as  $  C [ a, b] $
 +
or  $  L _ {p} ( a, b) $,  
 +
p \geq  1 $),
 +
and if
  
<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/p/p073/p073730/p0737309.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$
 +
U _ {n}  = \{ u _ {1} ( t) \dots u _ {n} ( t) \}
 +
$$
 +
 
 +
is a system of linearly independent functions in  $  X $,
 +
then for any  $  x \in X $
 +
the (generalized) polynomial of best approximation
 +
 
 +
$$ \tag{* }
 +
\widetilde{u}  ( t)  = \widetilde{u}  ( x, t)  = \
 +
\sum _ {k = 1 } ^ { n }  \widetilde{c}  _ {k} u _ {k} ( t),
 +
$$
  
 
defined by the relation
 
defined by the relation
  
<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/p/p073/p073730/p07373010.png" /></td> </tr></table>
+
$$
 +
\| x - \widetilde{u}  \|  = \
 +
\min _ {\{ c _ {k} \} }  \left \| x -
 +
\sum _ {k = 1 } ^ { n }  c _ {k} u _ {k} \right \| ,
 +
$$
  
exists. The polynomial of best approximation is unique for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373011.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373012.png" /> is a space with a strictly convex norm (i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373015.png" />). This is the case for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373017.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373018.png" />, which has a norm that is not strictly convex, the polynomial of best approximation for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373019.png" /> is unique if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373020.png" /> is a [[Chebyshev system|Chebyshev system]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373021.png" />, i.e. if each polynomial
+
exists. The polynomial of best approximation is unique for all $  x \in X $
 +
if $  X $
 +
is a space with a strictly convex norm (i.e. if $  \| x \| = \| y \| $
 +
and $  x \neq y $,  
 +
then $  \| x + y \| < \| x \| + \| y \| $).  
 +
This is the case for $  L _ {p} ( a, b) $,  
 +
$  1 < p < \infty $.  
 +
In $  C [ a, b] $,  
 +
which has a norm that is not strictly convex, the polynomial of best approximation for any $  x \in C [ a, b] $
 +
is unique if $  U _ {n} $
 +
is a [[Chebyshev system|Chebyshev system]] on $  [ a, b] $,  
 +
i.e. if each 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/p/p073/p073730/p07373022.png" /></td> </tr></table>
+
$$
 +
u ( t)  = \sum _ {k = 1 } ^ { n }  c _ {k} u _ {k} ( t)  \neq  0
 +
$$
  
has at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373023.png" /> zeros on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373024.png" />. In particular, one has uniqueness in the case of the (usual) algebraic polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373025.png" />, and also for the trigonometric polynomials in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373026.png" /> of continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373027.png" />-periodic functions on the real line, with the uniform metric. If the polynomial of best approximation exists and is unique for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373028.png" />, it is a continuous function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373029.png" />.
+
has at most $  n - 1 $
 +
zeros on $  [ a, b] $.  
 +
In particular, one has uniqueness in the case of the (usual) algebraic polynomials in $  C [ a, b] $,  
 +
and also for the trigonometric polynomials in the space $  C _ {2 \pi }  $
 +
of continuous $  2 \pi $-
 +
periodic functions on the real line, with the uniform metric. If the polynomial of best approximation exists and is unique for any $  x \in X $,  
 +
it is a continuous function of $  x $.
  
Necessary and sufficient conditions for a polynomial to be a best approximation in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373031.png" /> are known. For example, one has Chebyshev's theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373032.png" /> is a Chebyshev system, then the polynomial (*) is a polynomial of best approximation for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373033.png" /> in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373034.png" /> if and only if there exists a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373035.png" /> points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373037.png" />, at which the difference
+
Necessary and sufficient conditions for a polynomial to be a best approximation in the spaces $  C[ a, b] $
 +
and $  L _ {p} [ a, b] $
 +
are known. For example, one has Chebyshev's theorem: If $  U _ {n} $
 +
is a Chebyshev system, then the polynomial (*) is a polynomial of best approximation for a function $  x \in C [ a, b] $
 +
in the metric of $  C [ a, b] $
 +
if and only if there exists a system of $  n + 1 $
 +
points $  t _ {i} $,
 +
$  a \leq  t _ {1} < {} \dots < t _ {n + 1 }  \leq  b $,  
 +
at which the difference
  
<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/p/p073/p073730/p07373038.png" /></td> </tr></table>
+
$$
 +
\Delta ( t)  = x ( t) - \widetilde{u}  ( t)
 +
$$
  
 
assumes values
 
assumes values
  
<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/p/p073/p073730/p07373039.png" /></td> </tr></table>
+
$$
 +
\pm  \max _ {a \leq  t \leq  b }  | \Delta ( t) |
 +
$$
  
 
and, moreover,
 
and, moreover,
  
<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/p/p073/p073730/p07373040.png" /></td> </tr></table>
+
$$
 +
\Delta ( t _ {i + 1 }  )  = - \Delta ( t _ {i} ),\ \
 +
i = 1 \dots n.
 +
$$
  
The polynomial (*) is a polynomial of best approximation for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373042.png" />, in the metric of that space, if and only if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373043.png" />,
+
The polynomial (*) is a polynomial of best approximation for a function $  x ( t) \in L _ {p} [ a, b] $,  
 +
p > 1 $,  
 +
in the metric of that space, if and only if for $  k = 1 \dots n $,
  
<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/p/p073/p073730/p07373044.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { a } ^ { b }
 +
u _ {k} ( t)  | x ( t) - \widetilde{u}  ( t) | ^ {p - 1 } \
 +
\mathop{\rm sign}  [ x ( t) - \widetilde{u}  ( t)]  dt  = 0.
 +
$$
  
In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373045.png" />, the conditions
+
In $  L _ {1} [ a, b] $,
 +
the conditions
  
<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/p/p073/p073730/p07373046.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { a } ^ { b }
 +
u _ {k} ( t)  \mathop{\rm sign}  [ x ( t) - \widetilde{u}  ( t)]  dt  = 0,\ \
 +
k = 1 \dots n,
 +
$$
  
are sufficient for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373047.png" /> to be a polynomial of best approximation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373048.png" />, and if the measure of the set of all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373049.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373050.png" /> is zero, they are also necessary; see also [[Markov criterion|Markov criterion]].
+
are sufficient for $  \widetilde{u}  ( t) $
 +
to be a polynomial of best approximation for $  x \in L _ {1} [ a, b] $,  
 +
and if the measure of the set of all points $  t \in ( a, b) $
 +
at which $  x ( t) = \widetilde{u}  ( t) $
 +
is zero, they are also necessary; see also [[Markov criterion|Markov criterion]].
  
 
There exist algorithms for the approximate construction of polynomials of best uniform approximation (see e.g. [[#References|[3]]], [[#References|[5]]]).
 
There exist algorithms for the approximate construction of polynomials of best uniform approximation (see e.g. [[#References|[3]]], [[#References|[5]]]).
Line 43: Line 120:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,  "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.P. Korneichuk,  "Extremal problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.K. Dzyadyk,  "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.M. Tikhomirov,  "Some problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.J. Laurent,  "Approximation et optimisation" , Hermann  (1972)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.Ya. Remez,  "Foundations of numerical methods of Chebyshev approximation" , Kiev  (1969)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,  "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.P. Korneichuk,  "Extremal problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.K. Dzyadyk,  "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.M. Tikhomirov,  "Some problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.J. Laurent,  "Approximation et optimisation" , Hermann  (1972)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.Ya. Remez,  "Foundations of numerical methods of Chebyshev approximation" , Kiev  (1969)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.M. Pinkus,  "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373051.png" />-approximation" , Cambridge Univ. Press  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Schönhage,  "Approximationstheorie" , de Gruyter  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.A. Watson,  "Approximation theory and numerical methods" , Wiley  (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.W. Cheney,  "Introduction to approximation theory" , McGraw-Hill  (1966)  pp. Chapts. 4&amp;6</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.J.D. Powell,  "Approximation theory and methods" , Cambridge Univ. Press  (1981)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.R. Rice,  "The approximation of functions" , '''1. Linear theory''' , Addison-Wesley  (1964)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  T.J. Rivlin,  "An introduction to the approximation of functions" , Blaisdell  (1969)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.M. Pinkus,  "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073730/p07373051.png" />-approximation" , Cambridge Univ. Press  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Schönhage,  "Approximationstheorie" , de Gruyter  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.A. Watson,  "Approximation theory and numerical methods" , Wiley  (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.W. Cheney,  "Introduction to approximation theory" , McGraw-Hill  (1966)  pp. Chapts. 4&amp;6</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.J.D. Powell,  "Approximation theory and methods" , Cambridge Univ. Press  (1981)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.R. Rice,  "The approximation of functions" , '''1. Linear theory''' , Addison-Wesley  (1964)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  T.J. Rivlin,  "An introduction to the approximation of functions" , Blaisdell  (1969)</TD></TR></table>

Latest revision as of 08:06, 6 June 2020


A polynomial furnishing the best approximation of a function $ x ( t) $ in some metric, relative to all polynomials constructed from a given (finite) system of functions. If $ X $ is a normed linear function space (such as $ C [ a, b] $ or $ L _ {p} ( a, b) $, $ p \geq 1 $), and if

$$ U _ {n} = \{ u _ {1} ( t) \dots u _ {n} ( t) \} $$

is a system of linearly independent functions in $ X $, then for any $ x \in X $ the (generalized) polynomial of best approximation

$$ \tag{* } \widetilde{u} ( t) = \widetilde{u} ( x, t) = \ \sum _ {k = 1 } ^ { n } \widetilde{c} _ {k} u _ {k} ( t), $$

defined by the relation

$$ \| x - \widetilde{u} \| = \ \min _ {\{ c _ {k} \} } \left \| x - \sum _ {k = 1 } ^ { n } c _ {k} u _ {k} \right \| , $$

exists. The polynomial of best approximation is unique for all $ x \in X $ if $ X $ is a space with a strictly convex norm (i.e. if $ \| x \| = \| y \| $ and $ x \neq y $, then $ \| x + y \| < \| x \| + \| y \| $). This is the case for $ L _ {p} ( a, b) $, $ 1 < p < \infty $. In $ C [ a, b] $, which has a norm that is not strictly convex, the polynomial of best approximation for any $ x \in C [ a, b] $ is unique if $ U _ {n} $ is a Chebyshev system on $ [ a, b] $, i.e. if each polynomial

$$ u ( t) = \sum _ {k = 1 } ^ { n } c _ {k} u _ {k} ( t) \neq 0 $$

has at most $ n - 1 $ zeros on $ [ a, b] $. In particular, one has uniqueness in the case of the (usual) algebraic polynomials in $ C [ a, b] $, and also for the trigonometric polynomials in the space $ C _ {2 \pi } $ of continuous $ 2 \pi $- periodic functions on the real line, with the uniform metric. If the polynomial of best approximation exists and is unique for any $ x \in X $, it is a continuous function of $ x $.

Necessary and sufficient conditions for a polynomial to be a best approximation in the spaces $ C[ a, b] $ and $ L _ {p} [ a, b] $ are known. For example, one has Chebyshev's theorem: If $ U _ {n} $ is a Chebyshev system, then the polynomial (*) is a polynomial of best approximation for a function $ x \in C [ a, b] $ in the metric of $ C [ a, b] $ if and only if there exists a system of $ n + 1 $ points $ t _ {i} $, $ a \leq t _ {1} < {} \dots < t _ {n + 1 } \leq b $, at which the difference

$$ \Delta ( t) = x ( t) - \widetilde{u} ( t) $$

assumes values

$$ \pm \max _ {a \leq t \leq b } | \Delta ( t) | $$

and, moreover,

$$ \Delta ( t _ {i + 1 } ) = - \Delta ( t _ {i} ),\ \ i = 1 \dots n. $$

The polynomial (*) is a polynomial of best approximation for a function $ x ( t) \in L _ {p} [ a, b] $, $ p > 1 $, in the metric of that space, if and only if for $ k = 1 \dots n $,

$$ \int\limits _ { a } ^ { b } u _ {k} ( t) | x ( t) - \widetilde{u} ( t) | ^ {p - 1 } \ \mathop{\rm sign} [ x ( t) - \widetilde{u} ( t)] dt = 0. $$

In $ L _ {1} [ a, b] $, the conditions

$$ \int\limits _ { a } ^ { b } u _ {k} ( t) \mathop{\rm sign} [ x ( t) - \widetilde{u} ( t)] dt = 0,\ \ k = 1 \dots n, $$

are sufficient for $ \widetilde{u} ( t) $ to be a polynomial of best approximation for $ x \in L _ {1} [ a, b] $, and if the measure of the set of all points $ t \in ( a, b) $ at which $ x ( t) = \widetilde{u} ( t) $ is zero, they are also necessary; see also Markov criterion.

There exist algorithms for the approximate construction of polynomials of best uniform approximation (see e.g. [3], [5]).

References

[1] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[2] N.P. Korneichuk, "Extremal problems in approximation theory" , Moscow (1976) (In Russian)
[3] V.K. Dzyadyk, "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow (1977) (In Russian)
[4] V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)
[5] P.J. Laurent, "Approximation et optimisation" , Hermann (1972)
[6] E.Ya. Remez, "Foundations of numerical methods of Chebyshev approximation" , Kiev (1969) (In Russian)

Comments

References

[a1] A.M. Pinkus, "On -approximation" , Cambridge Univ. Press (1989)
[a2] A. Schönhage, "Approximationstheorie" , de Gruyter (1971)
[a3] G.A. Watson, "Approximation theory and numerical methods" , Wiley (1980)
[a4] E.W. Cheney, "Introduction to approximation theory" , McGraw-Hill (1966) pp. Chapts. 4&6
[a5] M.J.D. Powell, "Approximation theory and methods" , Cambridge Univ. Press (1981)
[a6] J.R. Rice, "The approximation of functions" , 1. Linear theory , Addison-Wesley (1964)
[a7] T.J. Rivlin, "An introduction to the approximation of functions" , Blaisdell (1969)
How to Cite This Entry:
Polynomial of best approximation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polynomial_of_best_approximation&oldid=48237
This article was adapted from an original article by N.P. KorneichukV.P. Motornyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article