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''finite trigonometric sum''
 
''finite trigonometric sum''
  
 
An expression of the form
 
An expression of 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/t094230/t0942301.png" /></td> </tr></table>
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$$
 +
T ( x)  = {
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\frac{a _ {0} }{2}
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} +
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\sum _ {k = 1 } ^ { n }  ( a _ {k}  \cos  kx + b _ {k}  \sin  kx)
 +
$$
  
with real coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094230/t0942302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094230/t0942303.png" />; the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094230/t0942304.png" /> is called the order of the trigonometric polynomial (provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094230/t0942305.png" />). A trigonometric polynomial can be written in complex form:
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with real coefficients $  a _ {0} , a _ {k} , b _ {k} $,  
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$  k = 1 \dots n $;  
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the number $  n $
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is called the order of the trigonometric polynomial (provided $  | a _ {n} | + | b _ {n} | > 0 $).  
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A trigonometric polynomial can be written in complex 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/t094230/t0942306.png" /></td> </tr></table>
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$$
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T ( x)  = \sum _ {k = - n } ^ { n }  c _ {k} e  ^ {ikx} ,
 +
$$
  
 
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/t094230/t0942307.png" /></td> </tr></table>
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$$
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2c _ {k}  = \left \{
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\begin{array}{ll}
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a _ {k} - ib _ {k} ,  &k \geq  0 \  ( \textrm{ with }  b _ {0} = 0),  \\
 +
a _ {-} k + ib _ {-} k ,  &k < 0 . \\
 +
\end{array}
 +
 
 +
\right .$$
  
 
Trigonometric polynomials are an important tool in the [[Approximation of functions|approximation of functions]].
 
Trigonometric polynomials are an important tool in the [[Approximation of functions|approximation of functions]].
 
 
  
 
====Comments====
 
====Comments====
 
Cf. also [[Trigonometric series|Trigonometric series]].
 
Cf. also [[Trigonometric series|Trigonometric series]].

Revision as of 14:56, 7 June 2020


finite trigonometric sum

An expression of the form

$$ T ( x) = { \frac{a _ {0} }{2} } + \sum _ {k = 1 } ^ { n } ( a _ {k} \cos kx + b _ {k} \sin kx) $$

with real coefficients $ a _ {0} , a _ {k} , b _ {k} $, $ k = 1 \dots n $; the number $ n $ is called the order of the trigonometric polynomial (provided $ | a _ {n} | + | b _ {n} | > 0 $). A trigonometric polynomial can be written in complex form:

$$ T ( x) = \sum _ {k = - n } ^ { n } c _ {k} e ^ {ikx} , $$

where

$$ 2c _ {k} = \left \{ \begin{array}{ll} a _ {k} - ib _ {k} , &k \geq 0 \ ( \textrm{ with } b _ {0} = 0), \\ a _ {-} k + ib _ {-} k , &k < 0 . \\ \end{array} \right .$$

Trigonometric polynomials are an important tool in the approximation of functions.

Comments

Cf. also Trigonometric series.

How to Cite This Entry:
Trigonometric polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trigonometric_polynomial&oldid=49479
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article