# Trigonometric polynomial

*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=49827