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, | ||
− | + | $$1,\cos x,\sin x,\dots,\cos nx,\sin nx,\dots,$$ | |
− | are orthogonal on any interval of the form | + | 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 | + | 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 | + | 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) |
Trigonometric system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trigonometric_system&oldid=19218