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Difference between revisions of "Approximation in the mean"

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Approximation of a given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a0129101.png" />, integrable on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a0129102.png" />, by a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a0129103.png" />, where the quantity
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Approximation of a given function $f(t)$, integrable on an interval $[a,b]$, by a function $\phi(t)$, where 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/a012910/a0129104.png" /></td> </tr></table>
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$$\mu(f,\phi)=\int\limits_a^b|f(t)-\phi(t)|dt$$
  
 
is taken as the measure of approximation.
 
is taken as the measure of approximation.
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The more general case, when
 
The more general case, when
  
<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/a012910/a0129105.png" /></td> </tr></table>
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$$\mu(f,\phi)=\int\limits_a^b|f(t)-\phi(t)|^qd\sigma(t)\quad(q>0),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a0129106.png" /> is a non-decreasing function different from a constant on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a0129107.png" />, is called mean-power approximation (with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a0129108.png" />) with respect to the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a0129109.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a01291010.png" /> is absolutely continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a01291011.png" />, then one obtains mean-power approximation with weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a01291012.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a01291013.png" /> is a step function with jumps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a01291014.png" /> at points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a01291015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a01291016.png" />, one has weighted mean-power approximation with respect to the system of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012910/a01291017.png" /> with measure of approximation
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where $\sigma(t)$ is a non-decreasing function different from a constant on $[a,b]$, is called mean-power approximation (with exponent $q$) with respect to the distribution $d\sigma(t)$. If $\sigma(t)$ is absolutely continuous and $\phi(t)=\sigma(t)$, then one obtains mean-power approximation with weight $\phi(t)$, and if $\sigma(t)$ is a step function with jumps $c_k$ at points $t_k$ in $[a,b]$, one has weighted mean-power approximation with respect to the system of points $\{t_k\}$ with measure of 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/a012910/a01291018.png" /></td> </tr></table>
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$$\mu(f,\phi)=\sum_kc_k|f(t_k)-\phi(t_k)|^q.$$
  
 
These concepts are extended in a natural way to the case of functions of several variables.
 
These concepts are extended in a natural way to the case of functions of several variables.

Latest revision as of 08:01, 23 August 2014

Approximation of a given function $f(t)$, integrable on an interval $[a,b]$, by a function $\phi(t)$, where the quantity

$$\mu(f,\phi)=\int\limits_a^b|f(t)-\phi(t)|dt$$

is taken as the measure of approximation.

The more general case, when

$$\mu(f,\phi)=\int\limits_a^b|f(t)-\phi(t)|^qd\sigma(t)\quad(q>0),$$

where $\sigma(t)$ is a non-decreasing function different from a constant on $[a,b]$, is called mean-power approximation (with exponent $q$) with respect to the distribution $d\sigma(t)$. If $\sigma(t)$ is absolutely continuous and $\phi(t)=\sigma(t)$, then one obtains mean-power approximation with weight $\phi(t)$, and if $\sigma(t)$ is a step function with jumps $c_k$ at points $t_k$ in $[a,b]$, one has weighted mean-power approximation with respect to the system of points $\{t_k\}$ with measure of approximation

$$\mu(f,\phi)=\sum_kc_k|f(t_k)-\phi(t_k)|^q.$$

These concepts are extended in a natural way to the case of functions of several variables.

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. Linear theory , Addison-Wesley (1964)


Comments

References

[a1] A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)
[a2] T.J. Rivlin, "An introduction to the approximation of functions" , Dover, reprint (1981)
[a3] E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff
How to Cite This Entry:
Approximation in the mean. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximation_in_the_mean&oldid=33102
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