# Trigonometric polynomial

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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.