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→‎See also: removed link to Godel's proof because the relation to it, not yet well pointed to in the article, is gratuitous. Removed reference to Many Worlds physics for the same reason. Removed reference to Borges, no significant connection.
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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 06:14, 11 February 2014

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