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Difference between revisions of "Trigonometric system"

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One of the most important orthogonal systems of functions (cf. [[Orthogonal system|Orthogonal system]]). The functions of the trigonometric system,
 
One of the most important orthogonal systems of functions (cf. [[Orthogonal system|Orthogonal system]]). The functions of the trigonometric system,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094270/t0942701.png" /></td> </tr></table>
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$$1,\cos x,\sin x,\dots,\cos nx,\sin nx,\dots,$$
  
are orthogonal on any interval of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094270/t0942702.png" />, while the functions
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are orthogonal on any interval of the form $[a-\pi,a+\pi]$, while the functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094270/t0942703.png" /></td> </tr></table>
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$$\frac{1}{\sqrt{2\pi}},\frac{\cos x}{\sqrt\pi},\frac{\sin x}{\sqrt\pi},\dots,\frac{\cos nx}{\sqrt\pi},\frac{\sin nx}{\sqrt\pi},\dots,$$
  
are orthonormal on this interval. The trigonometric system is complete and closed in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094270/t0942704.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094270/t0942705.png" />, and also in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094270/t0942706.png" /> of continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094270/t0942707.png" />-periodic functions. This system forms a [[Basis|basis]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094270/t0942708.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094270/t0942709.png" />. Series in the trigonometric system are studied in the theory of [[Trigonometric series|trigonometric series]].
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are orthonormal on this interval. The trigonometric system is complete and closed in the space $L_p[-\pi,\pi]$ for $1\leq p<\infty$, and also in the space $C_{2\pi}$ of continuous $2\pi$-periodic functions. This system forms a [[Basis|basis]] in $L_p[-\pi,\pi]$ for $1<p<\infty$. Series in the trigonometric system are studied in the theory of [[Trigonometric series|trigonometric series]].
  
Alongside the trigonometric system, wide use is made of the complex trigonometric system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094270/t09427010.png" />. The functions of these systems are related to one another by the [[Euler formulas|Euler formulas]].
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Alongside the trigonometric system, wide use is made of the complex trigonometric system $\{e^{inx}\}_{n=-\infty}^\infty$. The functions of these systems are related to one another by the [[Euler formulas|Euler formulas]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>

Latest revision as of 17:27, 26 September 2014

One of the most important orthogonal systems of functions (cf. Orthogonal system). The functions of the trigonometric system,

$$1,\cos x,\sin x,\dots,\cos nx,\sin nx,\dots,$$

are orthogonal on any interval of the form $[a-\pi,a+\pi]$, while the functions

$$\frac{1}{\sqrt{2\pi}},\frac{\cos x}{\sqrt\pi},\frac{\sin x}{\sqrt\pi},\dots,\frac{\cos nx}{\sqrt\pi},\frac{\sin nx}{\sqrt\pi},\dots,$$

are orthonormal on this interval. The trigonometric system is complete and closed in the space $L_p[-\pi,\pi]$ for $1\leq p<\infty$, and also in the space $C_{2\pi}$ of continuous $2\pi$-periodic functions. This system forms a basis in $L_p[-\pi,\pi]$ for $1<p<\infty$. Series in the trigonometric system are studied in the theory of trigonometric series.

Alongside the trigonometric system, wide use is made of the complex trigonometric system $\{e^{inx}\}_{n=-\infty}^\infty$. The functions of these systems are related to one another by the Euler formulas.

References

[1] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[2] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
How to Cite This Entry:
Trigonometric system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trigonometric_system&oldid=19218
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article