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The approximate representation of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t0942201.png" /> in the form of a [[Trigonometric polynomial|trigonometric polynomial]]
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{{TEX|done}}
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The approximate representation of a function $f$ in the form of a [[Trigonometric polynomial|trigonometric polynomial]]
  
<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/t/t094/t094220/t0942202.png" /></td> </tr></table>
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$$T(x)=A+\sum_{k=1}^n(a_k\cos kx+b_k\sin kx)$$
  
whose values coincide at prescribed points with the corresponding values of the function. Thus, it is always possible to choose the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t0942203.png" /> coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t0942204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t0942205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t0942206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t0942207.png" />, of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t0942208.png" />-th order polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t0942209.png" /> so that its values are equal to the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t09422010.png" /> of the function at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t09422011.png" /> preassigned points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t09422012.png" /> in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t09422013.png" />. The polynomial has the form
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whose values coincide at prescribed points with the corresponding values of the function. Thus, it is always possible to choose the $2n+1$ coefficients $A$, $a_k$, $b_k$, $k=1,\dots,n$, of the $n$-th order polynomial $T$ so that its values are equal to the values $y_k$ of the function at $2n+1$ preassigned points $x_k$ in the interval $[0,2\pi)$. The polynomial has the form
  
<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/t/t094/t094220/t09422014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$T(x)=\sum_{k=0}^{2n}y_kt_k(x),\tag{*}$$
  
 
where
 
where
  
<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/t/t094/t094220/t09422015.png" /></td> </tr></table>
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$$t_k(x)=\frac{\Delta x}{\Delta'(x)2\sin(x-x_k)/2},\quad\Delta(x)=\prod_{k=0}^{2n}2\sin\frac{x-x_k}{2}.$$
  
The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t09422016.png" /> assumes an especially simple form in case the nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t09422017.png" /> are equi-distant; the coefficients are given by the formulas
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The polynomial $T$ assumes an especially simple form in case the nodes $x_k=2k\pi/(2n+1)$ are equi-distant; the coefficients are given by the formulas
  
<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/t/t094/t094220/t09422018.png" /></td> </tr></table>
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$$A=\frac{1}{2n+1}\sum_{k=0}^{2n}y_k,$$
  
<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/t/t094/t094220/t09422019.png" /></td> </tr></table>
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$$a_m=\frac{2}{2n+1}\sum_{k=0}^{2n}y_k\cos mx_k,$$
  
<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/t/t094/t094220/t09422020.png" /></td> </tr></table>
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$$b_m=\frac{2}{2n+1}\sum_{k=0}^{2n}y_k\sin mx_k,\quad1\leq m\leq n.$$
  
  
  
 
====Comments====
 
====Comments====
The formula (*) above for the trigonometric polynomial taking the prescribed values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t09422021.png" /> at the nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094220/t09422022.png" /> is known as the Gauss formula of trigonometric interpolation, [[#References|[a2]]].
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The formula \ref{*} above for the trigonometric polynomial taking the prescribed values $y_k$ at the nodes $x_k$ is known as the Gauss formula of trigonometric interpolation, [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.J. Davis,  "Interpolation and approximation" , Dover, reprint  (1975)  pp. 29, 38</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.J. Davis,  "Interpolation and approximation" , Dover, reprint  (1975)  pp. 29, 38</TD></TR></table>

Revision as of 14:32, 3 June 2016

The approximate representation of a function $f$ in the form of a trigonometric polynomial

$$T(x)=A+\sum_{k=1}^n(a_k\cos kx+b_k\sin kx)$$

whose values coincide at prescribed points with the corresponding values of the function. Thus, it is always possible to choose the $2n+1$ coefficients $A$, $a_k$, $b_k$, $k=1,\dots,n$, of the $n$-th order polynomial $T$ so that its values are equal to the values $y_k$ of the function at $2n+1$ preassigned points $x_k$ in the interval $[0,2\pi)$. The polynomial has the form

$$T(x)=\sum_{k=0}^{2n}y_kt_k(x),\tag{*}$$

where

$$t_k(x)=\frac{\Delta x}{\Delta'(x)2\sin(x-x_k)/2},\quad\Delta(x)=\prod_{k=0}^{2n}2\sin\frac{x-x_k}{2}.$$

The polynomial $T$ assumes an especially simple form in case the nodes $x_k=2k\pi/(2n+1)$ are equi-distant; the coefficients are given by the formulas

$$A=\frac{1}{2n+1}\sum_{k=0}^{2n}y_k,$$

$$a_m=\frac{2}{2n+1}\sum_{k=0}^{2n}y_k\cos mx_k,$$

$$b_m=\frac{2}{2n+1}\sum_{k=0}^{2n}y_k\sin mx_k,\quad1\leq m\leq n.$$


Comments

The formula \ref{*} above for the trigonometric polynomial taking the prescribed values $y_k$ at the nodes $x_k$ is known as the Gauss formula of trigonometric interpolation, [a2].

References

[a1] A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)
[a2] P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 29, 38
How to Cite This Entry:
Trigonometric interpolation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trigonometric_interpolation&oldid=13422
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article