View source for Approximation of functions, measure of

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

<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>

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

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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]]).

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.

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299025.png" /> 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 <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]]:

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The quantity

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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.

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" />.

====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>



====Comments====
The measure of approximation is also called the error measure.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.G. Lorentz,   "Approximation of functions" , Holt, Rinehart &amp; Winston  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Pinkus,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012990/a01299047.png" />-widths in approximation theory" , Springer  (1985)  (Translated from Russian)</TD></TR></table>