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In the [[mathematical]] subfields of [[numerical analysis]] and [[mathematical analysis]], a '''trigonometric polynomial''' is a finite [[linear combination]] of [[Function (mathematics)|functions]] sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more [[natural number]]s. The coefficients may be taken as real numbers, for real-valued functions. For [[complex number|complex coefficients]], there is no difference between such a function and a finite [[Fourier series]].

Trigonometric polynomials are widely used, for example in [[trigonometric interpolation]] applied to the  [[interpolation]] of [[periodic function]]s. They are used also in the [[discrete Fourier transform]].

The term ''trigonometric polynomial'' for the real-valued case can be seen as using the [[analogy]]: the functions sin(''nx'') and cos(''nx'') are similar to the [[monomial basis]] for [[polynomial]]s. In the complex case the trigonometric polynomials are spanned by the positive and negative powers of ''e''<sup>''ix''</sup>.

==Formal definition==
Any function ''T'' of the form

:<math>T(x) = a_0 + \sum_{n=1}^N a_n \cos (nx) + i\sum_{n=1}^N b_n \sin(nx) \qquad (x \in \mathbb{R})</math>

with <math>a_n, b_n \in \mathbb{C}</math> for <math>0 \leq n \leq N</math>, is called a '''complex trigonometric polynomial''' of degree ''N'' {{harv|Rudin|1987|p=88}}. Using [[Euler's formula]] the polynomial can be rewritten as

:<math>T(x) = \sum_{n=-N}^N c_n e^{inx} \qquad (x \in \mathbb{R}).</math>

Analogously, letting <math>a_n,  b_n \in \mathbb{R}, \quad 0 \leq n \leq N</math> and <math>a_N \neq 0</math> or <math>b_N \neq 0</math>, then

:<math>t(x) = a_0 + \sum_{n=1}^N a_n \cos (nx) + \sum_{n=1}^N b_n \sin(nx) \qquad (x \in \mathbb{R})</math>

is called a '''real trigonometric polynomial''' of degree ''N'' {{harv|Powell|1981|p=150}}.

==Properties==

A trigonometric polynomial can be considered a [[periodic function]] on the [[real line]], with [[Periodic function|period]] some multiple of 2{{pi}}, or as a function on the [[unit circle]].

A basic result is that the trigonometric polynomials are [[dense set|dense]] in the space of [[continuous function]]s on the unit circle, with the [[uniform norm]] {{harv|Rudin|1987|loc=Thm 4.25}}; this is a special case of the [[Stone–Weierstrass theorem]]. More concretely, for every continuous function ''f'' and every ''&epsilon;'' > 0, there exists a trigonometric polynomial ''T'' such that |''f''(''z'') &minus; T(''z'')| < ''&epsilon;'' for all ''z''. [[Fejér's theorem]] states that the arithmetic means of the partial sums of the [[Fourier series]] of ''f'' converge uniformly to ''f'', provided ''f'' is continuous on the circle, thus giving an explicit way to find an approximating trigonometric polynomial ''T''.

A trigonometric polynomial of degree ''N'' has a maximum of 2''N'' roots in any open interval <nowiki>[</nowiki>''a'', ''a'' + 2{{pi}}<nowiki>)</nowiki> with a in '''R''', unless it is the zero function {{harv|Powell|1981|p=150}}.

==References==
* {{Citation | last1=Powell | first1=Michael J. D. | author1-link=Michael J. D. Powell | title=Approximation Theory and Methods | publisher=[[Cambridge University Press]] | isbn=978-0-521-29514-7 | year=1981}}
* {{Citation | last1=Rudin | first1=Walter | author1-link=Walter Rudin | title=Real and complex analysis | publisher=[[McGraw-Hill]] | location=New York | edition=3rd | isbn=978-0-07-054234-1 |mr=924157 | year=1987}}.

[[Category:Approximation theory]]
[[Category:Fourier analysis]]
[[Category:Polynomials]]
[[Category:Trigonometry]]