In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions. For complex coefficients, there is no difference between such a function and a finite Fourier series.
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 polynomials. In the complex case the trigonometric polynomials are spanned by the positive and negative powers of eix.
Any function T of the form
Analogously, letting an, bn be in R, 0 ≤ n ≤ N and aN ≠ 0 or bN ≠ 0, then
is called a real trigonometric polynomial of degree N (Powell 1981, p. 150).
A basic result is that the trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm (Rudin 1987, Thm 4.25); this is a special case of the Stone–Weierstrass theorem. More concretely, for every continuous function ƒ and every ε > 0, there exists a trigonometric polynomial T such that |ƒ(z) − T(z)| < ε for all z. Fejér's theorem states that the arithmetic means of the partial sums of the Fourier series of ƒ converge uniformly to ƒ, provided ƒ is continuous on the circle, thus giving an explicit way to find an approximating trigonometric polynomial T.
A trigonometric polynomial of degree N has a maximum of 2N roots in any open interval [a, a + 2π) with a in R, unless it is the zero function (Powell 1981, p. 150).