Trigonometric polynomial

<h2 id="definition">Definition</h2>
Any function T of the form

 
 
 
 T
 (
 x
 )
 =
 
 a
 
 0
 
 
 +
 
 ∑
 
 n
 =
 1
 
 
 N
 
 
 
 a
 
 n
 
 
 cos
 ⁡
 (
 n
 x
 )
 +
 
 ∑
 
 n
 =
 1
 
 
 N
 
 
 
 b
 
 n
 
 
 sin
 ⁡
 (
 n
 x
 )
 
 (
 x
 ∈
 
 R
 
 )
 
 
 {\displaystyle 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} )}

with coefficients 
 
 
 
 
 a
 
 n
 
 
 ,
 
 b
 
 n
 
 
 ∈
 
 C
 
 
 
 {\displaystyle a_{n},b_{n}\in \mathbb {C} }
 
 and at least one of the highest-degree coefficients 
 
 
 
 
 a
 
 N
 
 
 
 
 {\displaystyle a_{N}}
 
 and 
 
 
 
 
 b
 
 N
 
 
 
 
 {\displaystyle b_{N}}
 
 non-zero, is called a complex trigonometric polynomial of degree N.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> Using <a href="/facts/Euler%2527s_formula/Wq7VXIV0">Euler's formula</a> the polynomial can be rewritten as

 
 
 
 T
 (
 x
 )
 =
 
 ∑
 
 n
 =
 −
 N
 
 
 N
 
 
 
 c
 
 n
 
 
 
 e
 
 i
 n
 x
 
 
 
 (
 x
 ∈
 
 R
 
 )
 .
 
 
 {\displaystyle T(x)=\sum _{n=-N}^{N}c_{n}e^{inx}\qquad (x\in \mathbb {R} ).}

with 
 
 
 
 
 c
 
 n
 
 
 ∈
 
 C
 
 
 
 {\displaystyle c_{n}\in \mathbb {C} }
 
.
Analogously, letting coefficients 
 
 
 
 
 a
 
 n
 
 
 ,
 
 b
 
 n
 
 
 ∈
 
 R
 
 
 
 {\displaystyle a_{n},b_{n}\in \mathbb {R} }
 
, and at least one of 
 
 
 
 
 a
 
 N
 
 
 
 
 {\displaystyle a_{N}}
 
 and 
 
 
 
 
 b
 
 N
 
 
 
 
 {\displaystyle b_{N}}
 
 non-zero or, equivalently, 
 
 
 
 
 c
 
 n
 
 
 ∈
 
 R
 
 
 
 {\displaystyle c_{n}\in \mathbb {R} }
 
 and 
 
 
 
 
 c
 
 n
 
 
 =
 
 
 
 
 c
 ¯
 
 
 
 
 −
 n
 
 
 
 
 {\displaystyle c_{n}={\bar {c}}_{-n}}
 
 for all 
 
 
 
 n
 ∈
 [
 −
 N
 ,
 N
 ]
 
 
 {\displaystyle n\in [-N,N]}
 
, then

 
 
 
 t
 (
 x
 )
 =
 
 a
 
 0
 
 
 +
 
 ∑
 
 n
 =
 1
 
 
 N
 
 
 
 a
 
 n
 
 
 cos
 ⁡
 (
 n
 x
 )
 +
 
 ∑
 
 n
 =
 1
 
 
 N
 
 
 
 b
 
 n
 
 
 sin
 ⁡
 (
 n
 x
 )
 
 (
 x
 ∈
 
 R
 
 )
 
 
 {\displaystyle 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} )}

is called a real trigonometric polynomial of degree N.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="properties">Properties</h2>
A trigonometric polynomial can be considered a <a href="/facts/Periodic_function/Sm8lJQsF">periodic function</a> on the <a href="/facts/Real_line/6eq42xX5">real line</a>, with <a href="/facts/Periodic_function/Sm8lJQsF">period</a> some divisor of ⁠
 
 
 
 2
 π
 
 
 {\displaystyle 2\pi }
 
⁠, or as a function on the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a>.
Trigonometric polynomials are <a href="/facts/Dense_set/nPT9Yxao">dense</a> in the space of <a href="/facts/Continuous_function/sKbl02pB">continuous functions</a> on the unit circle, with the <a href="/facts/Uniform_norm/5yIkjNE3">uniform norm</a>;<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> this is a special case of the <a href="/facts/Stone%25E2%2580%2593Weierstrass_theorem/QB07Nwba">Stone–Weierstrass theorem</a>. More concretely, for every continuous function ⁠
 
 
 
 f
 
 
 {\displaystyle f}
 
⁠ and every ⁠
 
 
 
 ϵ
 >
 0
 
 
 {\displaystyle \epsilon >0}
 
⁠ there exists a trigonometric polynomial ⁠
 
 
 
 T
 
 
 {\displaystyle T}
 
⁠ such that 
 
 
 
 
 |
 
 f
 (
 z
 )
 −
 T
 (
 z
 )
 
 |
 
 <
 ϵ
 
 
 {\displaystyle |f(z)-T(z)|<\epsilon }
 
 for all ⁠
 
 
 
 z
 
 
 {\displaystyle z}
 
⁠. <a href="/facts/Fej%25C3%25A9r%2527s_theorem/8L8YqzHw">Fejér's theorem</a> states that the arithmetic means of the partial sums of the <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> of ⁠
 
 
 
 f
 
 
 {\displaystyle f}
 
⁠ converge uniformly to ⁠
 
 
 
 f
 
 
 {\displaystyle f}
 
⁠ provided ⁠
 
 
 
 f
 
 
 {\displaystyle f}
 
⁠ is continuous on the circle; these partial sums can be used to approximate ⁠
 
 
 
 f
 
 
 {\displaystyle f}
 
⁠.
A trigonometric polynomial of degree ⁠
 
 
 
 N
 
 
 {\displaystyle N}
 
⁠ has a maximum of ⁠
 
 
 
 2
 N
 
 
 {\displaystyle 2N}
 
⁠ roots in a real interval ⁠
 
 
 
 [
 a
 ,
 a
 +
 2
 π
 )
 
 
 {\displaystyle [a,a+2\pi )}
 
⁠ unless it is the zero function.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h2 id="fejr-riesz-theorem">Fejér-Riesz theorem</h2>
The Fejér-Riesz theorem states that every positive real trigonometric polynomial

t
 (
 x
 )
 =
 
 ∑
 
 n
 =
 −
 N
 
 
 N
 
 
 
 c
 
 n
 
 
 
 e
 
 i
 n
 x
 
 
 ,
 
 
 {\displaystyle t(x)=\sum _{n=-N}^{N}c_{n}e^{inx},}
 
 
satisfying 
 
 
 
 t
 (
 x
 )
 >
 0
 
 
 {\displaystyle t(x)>0}
 
 for all 
 
 
 
 x
 ∈
 
 R
 
 
 
 {\displaystyle x\in \mathbb {R} }
 
,
can be represented as the square of the <a href="/facts/Absolute_value/dwJBpEs5">modulus</a> of another (usually complex) trigonometric polynomial 
 
 
 
 q
 (
 x
 )
 
 
 {\displaystyle q(x)}
 
 such that:<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

t
        (
        x
        )
        =
        
          |
        
        q
        (
        x
        )
        
          
            |
          
          
            2
          
        
        =
        q
        (
        x
        )
        
          
            
              q
              ¯
            
          
        
        (
        x
        )
        .
      
    
    {\displaystyle t(x)=|q(x)|^{2}=q(x){\bar {q}}(x).}

Or, equivalently, every <a href="/facts/Laurent_polynomial/ESxjzdmw">Laurent polynomial</a>

w
        (
        z
        )
        =
        
          ∑
          
            n
            =
            −
            N
          
          
            N
          
        
        
          w
          
            n
          
        
        
          z
          
            n
          
        
        ,
      
    
    {\displaystyle w(z)=\sum _{n=-N}^{N}w_{n}z^{n},}

with 
  
    
      
        
          w
          
            n
          
        
        ∈
        
          C
        
      
    
    {\displaystyle w_{n}\in \mathbb {C} }
  
 that satisfies 
  
    
      
        w
        (
        ζ
        )
        ≥
        0
      
    
    {\displaystyle w(\zeta )\geq 0}
  
 for all 
  
    
      
        ζ
        ∈
        
          T
        
      
    
    {\displaystyle \zeta \in \mathbb {T} }
  
 can be written as:

w
        (
        ζ
        )
        =
        
          |
        
        p
        (
        ζ
        )
        
          
            |
          
          
            2
          
        
        =
        p
        (
        ζ
        )
        
          
            
              p
              ¯
            
          
        
        (
        
          
            
              ζ
              ¯
            
          
        
        )
        ,
      
    
    {\displaystyle w(\zeta )=|p(\zeta )|^{2}=p(\zeta ){\bar {p}}({\bar {\zeta }}),}

for some polynomial 
 
 
 
 p
 (
 z
 )
 
 
 {\displaystyle p(z)}
 
.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="notes">Notes</h2>

<ul><li>Dritschel, Michael A.; Rovnyak, James (2010). "The Operator Fejér-Riesz Theorem". A Glimpse at Hilbert Space Operators. Basel: Springer Basel. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-0346-0347-8_14">10.1007/978-3-0346-0347-8_14</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-0346-0346-1.</li>
<li>Hussen, Abdulmtalb; Zeyani, Abdelbaset (2021). "Fejer-Riesz Theorem and Its Generalization". International Journal of Scientific and Research Publications. 11 (6): 286–292. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.29322%2FIJSRP.11.06.2021.p11437">10.29322/IJSRP.11.06.2021.p11437</a>.</li>
<li><a href="/facts/Michael_J._D._Powell/rcNKU74P">Powell, Michael J. D.</a> (1981), Approximation Theory and Methods, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-29514-7</li>
<li>Riesz, Frigyes; Szőkefalvi-Nagy, Béla (1990). Functional analysis. New York: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-66289-3.</li>
<li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1987), Real and complex analysis (3rd ed.), New York: <a href="/facts/McGraw-Hill/oG8VpKJT">McGraw-Hill</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-054234-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0924157">0924157</a>.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Trigonometric_series/P6YD5EfS">Trigonometric series</a></li>
<li><a href="/facts/Quasi-polynomial/y9cCjlS1">Quasi-polynomial</a></li>
<li><a href="/facts/Exponential_polynomial/BEd1NfNM">Exponential polynomial</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Rudin 1987, p. 88 - Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 0924157 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0924157" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0924157</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Powell 1981, p. 150. - Powell, Michael J. D. (1981), Approximation Theory and Methods, Cambridge University Press, ISBN 978-0-521-29514-7 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Hussen & Zeyani 2021. - Hussen, Abdulmtalb; Zeyani, Abdelbaset (2021). "Fejer-Riesz Theorem and Its Generalization". International Journal of Scientific and Research Publications. 11 (6): 286–292. doi:10.29322/IJSRP.11.06.2021.p11437. <a href="https://doi.org/10.29322%2FIJSRP.11.06.2021.p11437" target="_blank">https://doi.org/10.29322%2FIJSRP.11.06.2021.p11437</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Rudin 1987, Thm 4.25 - Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 0924157 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0924157" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0924157</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Powell 1981, p. 150 - Powell, Michael J. D. (1981), Approximation Theory and Methods, Cambridge University Press, ISBN 978-0-521-29514-7 <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Riesz & Szőkefalvi-Nagy 1990, p. 117. - Riesz, Frigyes; Szőkefalvi-Nagy, Béla (1990). Functional analysis. New York: Dover Publications. ISBN 978-0-486-66289-3. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Dritschel & Rovnyak 2010, pp. 223–254. - Dritschel, Michael A.; Rovnyak, James (2010). "The Operator Fejér-Riesz Theorem". A Glimpse at Hilbert Space Operators. Basel: Springer Basel. doi:10.1007/978-3-0346-0347-8_14. ISBN 978-3-0346-0346-1. <a href="https://doi.org/10.1007%2F978-3-0346-0347-8_14" target="_blank">https://doi.org/10.1007%2F978-3-0346-0347-8_14</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Trigonometric polynomial open-in-new

Trigonometric polynomial