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.

Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are used also in the discrete Fourier transform.

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 ${\displaystyle e^{ix}}$, i.e., Laurent polynomials in ${\displaystyle z}$ under the change of variables ${\displaystyle x\mapsto z:=e^{ix}}$.

Any function T of the form

$${\displaystyle T(x)=a_{0}+\sum _{n=1}^{N}a_{n}\cos nx+\sum _{n=1}^{N}b_{n}\sin nx\qquad (x\in \mathbb {R} )}$$

with complex-valued coefficients ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$ and at least one of the highest-degree coefficients ${\displaystyle a_{N}}$ and ${\displaystyle b_{N}}$ non-zero, is called a complex trigonometric polynomial of degree N. The cosine and sine are the even and odd parts of the exponential of an imaginary variable, $${\displaystyle \cos nx={\tfrac {1}{2}}{\bigl (}e^{inx}+e^{-inx}{\bigr )},\quad \sin nx=-{\tfrac {1}{2}}i{\bigl (}e^{inx}-e^{-inx}{\bigr )},}$$ so the trigonometric polynomial can alternately be written as $${\displaystyle T(x)=\sum _{n=-N}^{N}c_{n}e^{inx}\qquad (x\in \mathbb {R} ),}$$ with complex coefficients ⁠${\displaystyle c_{0}=a_{0}}$⁠ and $${\displaystyle \quad c_{k}={\tfrac {1}{2}}(a_{k}-b_{k}i),\quad c_{-k}={\tfrac {1}{2}}(a_{k}+b_{k}i),}$$ for all ⁠${\displaystyle k}$⁠ from 1 to ⁠${\displaystyle N}$⁠.

If the coefficients ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$ are real for all ⁠${\displaystyle n}$⁠, then ⁠${\displaystyle T}$⁠ is called a real trigonometric polynomial. When using the exponential form, the complex coefficients satisfy ${\displaystyle c_{-n}={\overline {c}}_{n}}$ for all ${\displaystyle n\in [-N,N]}$.

A trigonometric polynomial can be considered a periodic function on the real line, with period some divisor of ⁠${\displaystyle 2\pi }$⁠, or as a function on the unit circle.