Editing Trigonometric polynomial

{{Short description|Concept in mathematics}}
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 <math>e^{ix}</math>, i.e., [[Laurent polynomial]]s in <math>z </math> under the [[change of variables]] <math>x \mapsto z := e^{ix}</math>.

==Definition==
Any function ''T'' of the form

<math display="block">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>

with coefficients <math>a_n, b_n \in \mathbb{C}</math> and at least one of the highest-degree coefficients <math>a_N</math> and <math>b_N</math> non-zero, is called a ''complex trigonometric polynomial'' of degree ''N''.<ref>{{harvnb|Rudin|1987|p=88}}</ref> Using [[Euler's formula]] the polynomial can be rewritten as

<math display="block">T(x) = \sum_{n=-N}^N c_n e^{inx} \qquad (x \in \mathbb{R}).</math>
with <math>c_{n}\in\mathbb{C}</math>.

Analogously, letting coefficients <math>a_n, b_n \in \mathbb{R}</math>, and at least one of <math>a_N</math> and <math>b_N</math> non-zero or, equivalently, <math>c_n \in \mathbb{R}</math> and <math>c_n = \bar{c}_{-n}</math> for all <math>n\in[-N,N]</math>, then

<math display="block">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''.{{sfn|Powell|1981|p=150}}{{sfn | Hussen | Zeyani | 2021}}

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

Trigonometric polynomials are [[dense set|dense]] in the space of [[continuous function]]s on the unit circle, with the [[uniform norm]];<ref>{{harvnb|Rudin|1987|loc=Thm 4.25}}</ref> this is a special case of the [[Stone–Weierstrass theorem]]. More concretely, for every continuous function {{tmath|f}} and every {{tmath|\epsilon > 0}} there exists a trigonometric polynomial {{tmath|T}} such that <math>|f(z) - T(z)| < \epsilon</math> for all {{tmath|z}}. [[Fejér's theorem]] states that the arithmetic means of the partial sums of the [[Fourier series]] of {{tmath|f}} converge uniformly to {{tmath|f}} provided {{tmath|f}} is continuous on the circle; these partial sums can be used to approximate {{tmath|f}}.

A trigonometric polynomial of degree {{tmath|N}} has a maximum of {{tmath|2N}} roots in a real interval {{tmath|[a, a+2\pi)}} unless it is the zero function.<ref>{{harvnb|Powell|1981|p=150}}</ref>

== Fejér-Riesz theorem ==
The Fejér-Riesz theorem states that every positive ''real'' trigonometric polynomial 
<math display="block">t(x) = \sum_{n=-N}^{N} c_n e^{i n x},</math> 
satisfying <math>t(x)>0</math> for all <math>x\in\mathbb{R}</math>,
can be represented as the square of the [[Absolute value#Complex numbers|modulus]] of another (usually ''complex'') trigonometric polynomial <math>q(x)</math> such that:{{sfn | Riesz | Szőkefalvi-Nagy | 1990 | p=117}}
<math display="block">t(x) = |q(x)|^2 = q(x)\bar{q}(x).</math>
Or, equivalently, every [[Laurent polynomial]]
<math display="block">w(z)=\sum_{n=-N}^{N} w_{n}z^{n},</math>
with <math>w_n \in\mathbb{C}</math> that satisfies <math>w(\zeta)\geq 0</math> for all <math>\zeta \in \mathbb{T}</math> can be written as:
<math display="block"> w(\zeta)=|p(\zeta)|^2=p(\zeta)\bar{p}(\bar{\zeta}),</math>
for some polynomial <math>p(z)</math>.{{sfn | Dritschel | Rovnyak | 2010 | pp=223–254}}

==Notes==
{{reflist}}

==References==
* {{cite book | last=Dritschel | first=Michael A. | last2=Rovnyak | first2=James | title=A Glimpse at Hilbert Space Operators | chapter=The Operator Fejér-Riesz Theorem | publisher=Springer Basel | publication-place=Basel | date=2010 | isbn=978-3-0346-0346-1 | doi=10.1007/978-3-0346-0347-8_14}}
* {{cite journal | last=Hussen | first=Abdulmtalb | last2=Zeyani | first2=Abdelbaset | title=Fejer-Riesz Theorem and Its Generalization | journal=International Journal of Scientific and Research Publications  | volume=11 | issue=6 | date=2021 | doi=10.29322/IJSRP.11.06.2021.p11437 | pages=286–292}}
* {{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}}
* {{cite book | last=Riesz | first=Frigyes | last2=Szőkefalvi-Nagy | first2=Béla | title=Functional analysis | publisher=Dover Publications | publication-place=New York | date=1990 | isbn=978-0-486-66289-3}}
* {{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}}.

==See also==
* [[Trigonometric series]]
* [[Quasi-polynomial]]
* [[Exponential polynomial]]

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