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A quantitative expression for the error of an approximation. When the discussion is about the approximation of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a0129901.png" /> by a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a0129902.png" />, the measure of approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a0129903.png" /> is usually defined by the metric in a function space containing both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a0129904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a0129905.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a0129906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a0129907.png" /> are continuous functions on a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a0129908.png" />, the uniform metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a0129909.png" /> is commonly used, i.e. one puts
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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/a/a012/a012990/a01299010.png" /></td> </tr></table>
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If continuity of the approximated function is not guaranteed or if the conditions of the problem imply that it is important that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299012.png" /> are close on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299013.png" /> in an average sense, the integral metric of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299014.png" /> may be used, putting
+
A quantitative expression for the error of an approximation. When the discussion is about the approximation of a function  $  f $
 +
by a function  $  \phi $,
 +
the measure of approximation  $  \mu (f, \phi ) $
 +
is usually defined by the metric in a function space containing both  $  f $
 +
and  $  \phi $.  
 +
For example, if  $  f $
 +
and $  \phi $
 +
are continuous functions on a segment  $  [a, b] $,  
 +
the uniform metric of $  C [a, b] $
 +
is commonly used, i.e. one puts
  
<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/a/a012/a012990/a01299015.png" /></td> </tr></table>
+
$$
 +
\mu (f, \phi )  = \
 +
\max _ {a \leq  t \leq  b } \
 +
| f (t) - \phi (t) | .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299016.png" /> is a weight function. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299017.png" /> is most often used and is most convenient from a practical point of view (cf. [[Mean-square approximation of a function|Mean-square approximation of a function]]).
+
If continuity of the approximated function is not guaranteed or if the conditions of the problem imply that it is important that  $  f $
 +
and $  \phi $
 +
are close on  $  [a, b] $
 +
in an average sense, the integral metric of a space  $  L _ {p} [a, b] $
 +
may be used, putting
  
The measure of approximation may take into account only values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299019.png" /> in discrete points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299021.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299022.png" />, e.g.
+
$$
 +
\mu (f, \phi )  = \
 +
\int\limits _ { a } ^ { b }
 +
q (t) | f (t) -
 +
\phi (t) |  ^ {p} \
 +
dt,\  p > 0,
 +
$$
  
<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/a/a012/a012990/a01299023.png" /></td> </tr></table>
+
where  $  q (t) $
 +
is a weight function. The case  $  p = 2 $
 +
is most often used and is most convenient from a practical point of view (cf. [[Mean-square approximation of a function|Mean-square approximation of a function]]).
  
<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/a/a012/a012990/a01299024.png" /></td> </tr></table>
+
The measure of approximation may take into account only values of  $  f $
 +
and  $  \phi $
 +
in discrete points  $  t _ {k} $,
 +
$  k = 1 \dots n $,
 +
of  $  [a, b] $,
 +
e.g.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299025.png" /> are certain positive coefficients.
+
$$
 +
\mu (f, \phi )  = \
 +
\max _ {1 \leq  k \leq  n } \
 +
| f (t _ {k} ) - \phi (t _ {k} ) | ,
 +
$$
 +
 
 +
$$
 +
\mu (f, \phi )  = \sum _ {k = 1 } ^ { n }  q _ {k} | f (t _ {k} ) - \phi (t _ {k} ) |  ^ {p} ,
 +
$$
 +
 
 +
where  $  q _ {k} $
 +
are certain positive coefficients.
  
 
One defines in an analogous way the measure of approximation of functions in two or more variables.
 
One defines in an analogous way the measure of approximation of functions in two or more variables.
  
The measure of approximation of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299026.png" /> by a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299027.png" /> of functions is usually taken to be the [[Best approximation|best approximation]]:
+
The measure of approximation of a function $  f $
 +
by a family $  F $
 +
of functions is usually taken to be the [[Best approximation|best approximation]]:
  
<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/a/a012/a012990/a01299028.png" /></td> </tr></table>
+
$$
 +
E (f, F)  = \
 +
\mu (f, F)  = \
 +
\inf _ {\phi \in F } \
 +
\mu (f, \phi ).
 +
$$
  
 
The quantity
 
The quantity
  
<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/a/a012/a012990/a01299029.png" /></td> </tr></table>
+
$$
 +
E ( \mathfrak M , F)  = \
 +
\mu ( \mathfrak M , F)  = \
 +
\sup _ {f \in \mathfrak M } \
 +
\inf _ {\phi \in F } \
 +
\mu (f, \phi )
 +
$$
  
is usually taken as the measure of approximation of a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299030.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299031.png" /> by functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299032.png" /> from a certain fixed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299033.png" />. It characterizes the maximal deviation of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299034.png" /> from functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299035.png" /> that are closest to them.
+
is usually taken as the measure of approximation of a class $  \mathfrak M $
 +
of functions $  f $
 +
by functions $  \phi $
 +
from a certain fixed set $  F $.  
 +
It characterizes the maximal deviation of functions in $  \mathfrak M $
 +
from functions in $  F $
 +
that are closest to them.
  
In general, when approximation in an arbitrary metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299036.png" /> is considered, the measure of approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299037.png" /> of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299038.png" /> by an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299039.png" /> (a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299040.png" />) is the distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299041.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299042.png" />) between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299044.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299045.png" />) in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299046.png" />.
+
In general, when approximation in an arbitrary metric space $  X $
 +
is considered, the measure of approximation $  \mu (x, u) $
 +
of an element $  x $
 +
by an element $  u $(
 +
a set $  F $)  
 +
is the distance $  \rho (x, u) $(
 +
or $  \rho (x, F) $)  
 +
between $  x $
 +
and $  u $(
 +
or $  F $)  
 +
in the metric of $  X $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.R. Rice,  "The approximation of functions" , '''1–2''' , Addison-Wesley  (1964–1968)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.R. Rice,  "The approximation of functions" , '''1–2''' , Addison-Wesley  (1964–1968)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 18:47, 5 April 2020


A quantitative expression for the error of an approximation. When the discussion is about the approximation of a function $ f $ by a function $ \phi $, the measure of approximation $ \mu (f, \phi ) $ is usually defined by the metric in a function space containing both $ f $ and $ \phi $. For example, if $ f $ and $ \phi $ are continuous functions on a segment $ [a, b] $, the uniform metric of $ C [a, b] $ is commonly used, i.e. one puts

$$ \mu (f, \phi ) = \ \max _ {a \leq t \leq b } \ | f (t) - \phi (t) | . $$

If continuity of the approximated function is not guaranteed or if the conditions of the problem imply that it is important that $ f $ and $ \phi $ are close on $ [a, b] $ in an average sense, the integral metric of a space $ L _ {p} [a, b] $ may be used, putting

$$ \mu (f, \phi ) = \ \int\limits _ { a } ^ { b } q (t) | f (t) - \phi (t) | ^ {p} \ dt,\ p > 0, $$

where $ q (t) $ is a weight function. The case $ p = 2 $ is most often used and is most convenient from a practical point of view (cf. Mean-square approximation of a function).

The measure of approximation may take into account only values of $ f $ and $ \phi $ in discrete points $ t _ {k} $, $ k = 1 \dots n $, of $ [a, b] $, e.g.

$$ \mu (f, \phi ) = \ \max _ {1 \leq k \leq n } \ | f (t _ {k} ) - \phi (t _ {k} ) | , $$

$$ \mu (f, \phi ) = \sum _ {k = 1 } ^ { n } q _ {k} | f (t _ {k} ) - \phi (t _ {k} ) | ^ {p} , $$

where $ q _ {k} $ are certain positive coefficients.

One defines in an analogous way the measure of approximation of functions in two or more variables.

The measure of approximation of a function $ f $ by a family $ F $ of functions is usually taken to be the best approximation:

$$ E (f, F) = \ \mu (f, F) = \ \inf _ {\phi \in F } \ \mu (f, \phi ). $$

The quantity

$$ E ( \mathfrak M , F) = \ \mu ( \mathfrak M , F) = \ \sup _ {f \in \mathfrak M } \ \inf _ {\phi \in F } \ \mu (f, \phi ) $$

is usually taken as the measure of approximation of a class $ \mathfrak M $ of functions $ f $ by functions $ \phi $ from a certain fixed set $ F $. It characterizes the maximal deviation of functions in $ \mathfrak M $ from functions in $ F $ that are closest to them.

In general, when approximation in an arbitrary metric space $ X $ is considered, the measure of approximation $ \mu (x, u) $ of an element $ x $ by an element $ u $( a set $ F $) is the distance $ \rho (x, u) $( or $ \rho (x, F) $) between $ x $ and $ u $( or $ F $) in the metric of $ X $.

References

[1] V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian)
[2] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[3] J.R. Rice, "The approximation of functions" , 1–2 , Addison-Wesley (1964–1968)

Comments

The measure of approximation is also called the error measure.

References

[a1] G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966)
[a2] A. Pinkus, "-widths in approximation theory" , Springer (1985) (Translated from Russian)
How to Cite This Entry:
Approximation of functions, measure of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximation_of_functions,_measure_of&oldid=45204
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