Trigonometric system

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)