Namespaces
Variants
Actions

Difference between revisions of "Best approximations, sequence of"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159102.png" /> of numbers, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159103.png" /> is the [[Best approximation|best approximation]] of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159104.png" /> of a normed linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159105.png" /> by elements of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159106.png" />-dimensional subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159107.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159108.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b0159109.png" />. Usually, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591010.png" /> is the linear span of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591011.png" /> elements of some fixed system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591012.png" /> of linearly independent elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591013.png" />.
+
<!--
 +
b0159101.png
 +
$#A+1 = 44 n = 0
 +
$#C+1 = 44 : ~/encyclopedia/old_files/data/B015/B.0105910 Best approximations, sequence of
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591015.png" /> is the subspace of algebraic polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591016.png" />, sequences of best approximations were first considered in the 1850s by P.L. Chebyshev; the fact that
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/b/b015/b015910/b01591017.png" /></td> </tr></table>
+
A sequence  $  \{ E (x, F _ {n} ) \} $,
 +
$  n = 1, 2 \dots $
 +
of numbers, where  $  E (x, F _ {n} ) $
 +
is the [[Best approximation|best approximation]] of an element  $  x $
 +
of a normed linear space  $  X $
 +
by elements of an  $  n $-
 +
dimensional subspace  $  F _ {n} \subset  X $,
 +
with  $  F _ {1} \subset  F _ {2} \subset  \dots $,
 +
so that  $  E (x, F _ {1} ) \geq  E (x, F _ {2} ) \geq  \dots $.
 +
Usually,  $  F _ {n} $
 +
is the linear span of the first  $  n $
 +
elements of some fixed system  $  \{ u _ {1} , u _ {2} , .  . . \} $
 +
of linearly independent elements of  $  X $.
  
for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591018.png" /> was established in 1885 by K. Weierstrass. In the general case, the relation
+
In the case  $  X = C [a, b] $
 +
and  $  F _ {n} = F _ {n}  ^ {A} $
 +
is the subspace of algebraic polynomials of degree  $  n - 1 $,
 +
sequences of best approximations were first considered in the 1850s by P.L. Chebyshev; the fact that
  
<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/b/b015/b015910/b01591019.png" /></td> </tr></table>
+
$$
 +
E (x, F _ {n}  ^ {A} )  \rightarrow  0 \  \textrm{ as }  n \rightarrow \infty
 +
$$
  
is always satisfied when the union of the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591021.png" /> is everywhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591022.png" />,
+
for any function  $  x (t) \in C [a, b] $
 +
was established in 1885 by K. Weierstrass. In the general case, 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/b/b015/b015910/b01591023.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {n \rightarrow \infty }  E (x, F _ {n} )  = 0 \ \
 +
\textrm{ for }  \textrm{ all }  x \in X
 +
$$
  
(essentially, this is an equivalent statement). However, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591024.png" /> may converge to zero arbitrarily slowly. This follows from a theorem of Bernstein: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591025.png" /> is a sequence of subspaces of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591026.png" /> of a normed linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591027.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591029.png" />, then, for any monotone decreasing null sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591030.png" /> of non-negative real numbers, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591033.png" />. In the function spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591035.png" />, the rate at which a sequence of best approximations tends to zero depends both on the system of subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591036.png" /> and on the smoothness characteristics of the approximated function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591037.png" /> (the modulus of continuity, the existence of derivatives up to a specific order, etc.), and it can be estimated in terms of these characteristics. Conversely, knowing the rate of convergence to zero of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591038.png" />, one can draw conclusions with respect to the smoothness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591039.png" /> (see [[Approximation of functions, direct and inverse theorems|Approximation of functions, direct and inverse theorems]]).
+
is always satisfied when the union of the subspaces  $  F _ {n} $,
 +
$  n = 1, 2 \dots $
 +
is everywhere dense in  $  X $,
 +
 
 +
$$
 +
\overline{ {\cup F _ {n} }}\;  =  X
 +
$$
 +
 
 +
(essentially, this is an equivalent statement). However, the sequence $  \{ E (x, F _ {n} ) \} $
 +
may converge to zero arbitrarily slowly. This follows from a theorem of Bernstein: If $  \{ F _ {n} \} $
 +
is a sequence of subspaces of dimension $  n = 1, 2 \dots $
 +
of a normed linear space $  X $,  
 +
such that $  F _ {1} \subset  F _ {2} \subset  \dots $
 +
and $  \overline{ {\cup F _ {n} }}\; = X $,  
 +
then, for any monotone decreasing null sequence $  \{ \mu _ {n} \} $
 +
of non-negative real numbers, there exists an $  x \in X $
 +
such that $  E (x, F _ {n} ) = \mu _ {n} $,
 +
$  n = 1, 2 ,\dots $.  
 +
In the function spaces $  C $
 +
and $  L _ {p} $,  
 +
the rate at which a sequence of best approximations tends to zero depends both on the system of subspaces $  F _ {n} $
 +
and on the smoothness characteristics of the approximated function $  x $(
 +
the modulus of continuity, the existence of derivatives up to a specific order, etc.), and it can be estimated in terms of these characteristics. Conversely, knowing the rate of convergence to zero of the sequence $  \{ E (x, F _ {n} ) \} $,  
 +
one can draw conclusions with respect to the smoothness of $  x (t) $(
 +
see [[Approximation of functions, direct and inverse theorems|Approximation of functions, direct and inverse theorems]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.N. Bernshtein,  "Collected works" , '''2''' , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.L. Goncharov,  "The theory of interpolation and approximation of functions" , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.F. Timan,  "Theory of approximation of functions of a real variable" , Pergamon  (1963)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.N. Bernshtein,  "Collected works" , '''2''' , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.L. Goncharov,  "The theory of interpolation and approximation of functions" , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.F. Timan,  "Theory of approximation of functions of a real variable" , Pergamon  (1963)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Theorems inferring smoothness characteristics of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591040.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591041.png" /> from properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591042.png" /> were first given by D. Jackson in 1911 for algebraic or trigonometric approximation, see [[Jackson theorem|Jackson theorem]]. Theorems converse to these, i.e. inferring properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591043.png" /> from smoothness characteristics of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015910/b01591044.png" />, have been proved by S.N. Bernstein [S.N. Bernshtein] and A. Zygmund, cf. [[Bernstein theorem|Bernstein theorem]]. See also [[#References|[a2]]], Chapt. 4, Sect. 6 and Chapt. 6, Sect. 3.
+
Theorems inferring smoothness characteristics of a function $  x \in C $
 +
or $  L _ {p} $
 +
from properties of $  E (x, F _ {n} ) $
 +
were first given by D. Jackson in 1911 for algebraic or trigonometric approximation, see [[Jackson theorem|Jackson theorem]]. Theorems converse to these, i.e. inferring properties of $  E (x, F _ {n} ) $
 +
from smoothness characteristics of the function $  x $,  
 +
have been proved by S.N. Bernstein [S.N. Bernshtein] and A. Zygmund, cf. [[Bernstein theorem|Bernstein theorem]]. See also [[#References|[a2]]], Chapt. 4, Sect. 6 and Chapt. 6, Sect. 3.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.P. Natanson,  "Constructive function theory" , '''1–3''' , F. Ungar  (1964–1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.W. Cheney,  "Introduction to approximation theory" , Chelsea, reprint  (1982)  pp. 203ff</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.P. Natanson,  "Constructive function theory" , '''1–3''' , F. Ungar  (1964–1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.W. Cheney,  "Introduction to approximation theory" , Chelsea, reprint  (1982)  pp. 203ff</TD></TR></table>

Revision as of 10:58, 29 May 2020


A sequence $ \{ E (x, F _ {n} ) \} $, $ n = 1, 2 \dots $ of numbers, where $ E (x, F _ {n} ) $ is the best approximation of an element $ x $ of a normed linear space $ X $ by elements of an $ n $- dimensional subspace $ F _ {n} \subset X $, with $ F _ {1} \subset F _ {2} \subset \dots $, so that $ E (x, F _ {1} ) \geq E (x, F _ {2} ) \geq \dots $. Usually, $ F _ {n} $ is the linear span of the first $ n $ elements of some fixed system $ \{ u _ {1} , u _ {2} , . . . \} $ of linearly independent elements of $ X $.

In the case $ X = C [a, b] $ and $ F _ {n} = F _ {n} ^ {A} $ is the subspace of algebraic polynomials of degree $ n - 1 $, sequences of best approximations were first considered in the 1850s by P.L. Chebyshev; the fact that

$$ E (x, F _ {n} ^ {A} ) \rightarrow 0 \ \textrm{ as } n \rightarrow \infty $$

for any function $ x (t) \in C [a, b] $ was established in 1885 by K. Weierstrass. In the general case, the relation

$$ \lim\limits _ {n \rightarrow \infty } E (x, F _ {n} ) = 0 \ \ \textrm{ for } \textrm{ all } x \in X $$

is always satisfied when the union of the subspaces $ F _ {n} $, $ n = 1, 2 \dots $ is everywhere dense in $ X $,

$$ \overline{ {\cup F _ {n} }}\; = X $$

(essentially, this is an equivalent statement). However, the sequence $ \{ E (x, F _ {n} ) \} $ may converge to zero arbitrarily slowly. This follows from a theorem of Bernstein: If $ \{ F _ {n} \} $ is a sequence of subspaces of dimension $ n = 1, 2 \dots $ of a normed linear space $ X $, such that $ F _ {1} \subset F _ {2} \subset \dots $ and $ \overline{ {\cup F _ {n} }}\; = X $, then, for any monotone decreasing null sequence $ \{ \mu _ {n} \} $ of non-negative real numbers, there exists an $ x \in X $ such that $ E (x, F _ {n} ) = \mu _ {n} $, $ n = 1, 2 ,\dots $. In the function spaces $ C $ and $ L _ {p} $, the rate at which a sequence of best approximations tends to zero depends both on the system of subspaces $ F _ {n} $ and on the smoothness characteristics of the approximated function $ x $( the modulus of continuity, the existence of derivatives up to a specific order, etc.), and it can be estimated in terms of these characteristics. Conversely, knowing the rate of convergence to zero of the sequence $ \{ E (x, F _ {n} ) \} $, one can draw conclusions with respect to the smoothness of $ x (t) $( see Approximation of functions, direct and inverse theorems).

References

[1] S.N. Bernshtein, "Collected works" , 2 , Moscow (1954) (In Russian)
[2] V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian)
[3] A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)

Comments

Theorems inferring smoothness characteristics of a function $ x \in C $ or $ L _ {p} $ from properties of $ E (x, F _ {n} ) $ were first given by D. Jackson in 1911 for algebraic or trigonometric approximation, see Jackson theorem. Theorems converse to these, i.e. inferring properties of $ E (x, F _ {n} ) $ from smoothness characteristics of the function $ x $, have been proved by S.N. Bernstein [S.N. Bernshtein] and A. Zygmund, cf. Bernstein theorem. See also [a2], Chapt. 4, Sect. 6 and Chapt. 6, Sect. 3.

References

[a1] I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian)
[a2] E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff
How to Cite This Entry:
Best approximations, sequence of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Best_approximations,_sequence_of&oldid=46040
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