View source for Trigonometric polynomial

''finite trigonometric sum''

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>

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:

<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>

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>

Trigonometric polynomials are an important tool in the [[Approximation of functions|approximation of functions]].



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