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Difference between revisions of "Deviation of an approximating function"

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The distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031440/d0314401.png" /> between the approximating function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031440/d0314402.png" /> and a given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031440/d0314403.png" />. In one and the same class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031440/d0314404.png" /> different metrics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031440/d0314405.png" /> may be considered, e.g. the uniform metric
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The distance $\rho(g,f)$ between the approximating function $g\in K$ and a given function $f\in\mathfrak M$. In one and the same class $\mathfrak M$ different metrics $\rho$ may be considered, e.g. the uniform metric
  
<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/d/d031/d031440/d0314406.png" /></td> </tr></table>
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$$\rho(g,f)=\max_{a\leq x\leq b}|g(x)-f(x)|,$$
  
 
an integral metric
 
an integral metric
  
<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/d/d031/d031440/d0314407.png" /></td> </tr></table>
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$$\rho(g,f)=\left(\int\limits_a^b|g(x)-f(x)|^pdx\right)^{1/p},\quad p\geq1,$$
  
and other metrics. As the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031440/d0314408.png" /> of approximating functions one may consider algebraic polynomials, trigonometric polynomials and also partial sums of orthogonal expansions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031440/d0314409.png" /> in an orthogonal system, linear averages of these partial sums as well as a number of other sets.
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and other metrics. As the class $K$ of approximating functions one may consider algebraic polynomials, trigonometric polynomials and also partial sums of orthogonal expansions of $f$ in an orthogonal system, linear averages of these partial sums as well as a number of other sets.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,  "Complete collected works" , '''2''' , Moscow-Leningrad  (1947)  (In Russian)</TD></TR><TR><TD valign="top">[2]</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">[3]</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">[4]</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">[5]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,  "Complete collected works" , '''2''' , Moscow-Leningrad  (1947)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Natanson,  "Constructive function theory" , '''1–3''' , F. Ungar  (1964–1965)  (Translated from Russian)</TD></TR>
 
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<TR><TD valign="top">[3]</TD> <TD valign="top">  V.L. Goncharov,  "The theory of interpolation and approximation of functions" , Moscow  (1954)  (In Russian)</TD></TR>
 
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<TR><TD valign="top">[4]</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">[5]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR>
====Comments====
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,  "Introduction to approximation theory" , Chelsea, reprint  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Schönhage,  "Approximationstheorie" , de Gruyter  (1971)</TD></TR>
 
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</table>
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,  "Introduction to approximation theory" , Chelsea, reprint  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Schönhage,  "Approximationstheorie" , de Gruyter  (1971)</TD></TR></table>
 

Latest revision as of 18:41, 11 April 2023

The distance $\rho(g,f)$ between the approximating function $g\in K$ and a given function $f\in\mathfrak M$. In one and the same class $\mathfrak M$ different metrics $\rho$ may be considered, e.g. the uniform metric

$$\rho(g,f)=\max_{a\leq x\leq b}|g(x)-f(x)|,$$

an integral metric

$$\rho(g,f)=\left(\int\limits_a^b|g(x)-f(x)|^pdx\right)^{1/p},\quad p\geq1,$$

and other metrics. As the class $K$ of approximating functions one may consider algebraic polynomials, trigonometric polynomials and also partial sums of orthogonal expansions of $f$ in an orthogonal system, linear averages of these partial sums as well as a number of other sets.

References

[1] P.L. Chebyshev, "Complete collected works" , 2 , Moscow-Leningrad (1947) (In Russian)
[2] I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian)
[3] V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian)
[4] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[5] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[a1] E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982)
[a2] A. Schönhage, "Approximationstheorie" , de Gruyter (1971)
How to Cite This Entry:
Deviation of an approximating function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Deviation_of_an_approximating_function&oldid=17554
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article