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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]].
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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]].
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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) + \mathrm{i}\sum_{n=1}^N b_n \sin(nx) \qquad (x \in \mathbf{R})</math>
 
with ''a''<sub>''n''</sub>, ''b''<sub>''n''</sub> in '''C''' for 0 ≤ ''n'' ≤ ''N'', 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 \mathrm{e}^{\mathrm{i}nx} \qquad (x \in \mathbf{R}).</math>
 
Analogously let ''a''<sub>''n''</sub>, ''b''<sub>''n''</sub> be in '''R''', 0 ≤ ''n'' ≤ ''N'' and ''a''<sub>''N''</sub> ≠ 0 or ''b''<sub>''N''</sub> ≠ 0 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 \mathbf{R})</math>
 
is called '''real trigonometric polynomial''' of degree ''N'' {{harv|Powell|1981|p=150}}.
 
==Notes==
 
A trigonometric polynomial can be considered a [[periodic function]] on the [[real line]], with [[Periodic function|period]] some multiple of 2π, 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 &fnof; and every &epsilon; > 0, there exists a trigonometric polynomial ''T'' such that |&fnof;(''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 &fnof; converge uniformly to &fnof;, 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'' + 2π) 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 | id={{MathSciNet | id = 924157}} | year=1987}}.
 
[[Category:Approximation theory]]
[[Category:Fourier analysis]]
[[Category:Polynomials]]
[[Category:Trigonometry]]

Revision as of 05:14, 11 February 2014

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