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''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|approximation of functions]].

====Comments====
Cf. also [[Trigonometric series|Trigonometric series]].