In the mathematical subfield of numerical analysis, a **trigonometric polynomial** is a finite linear linear combination of sin(*nx*) and cos(*nx*) with *n* a natural number. Hence the term trigonometric polynomial as the sin(*nx*)s and cos(*nx*)s are used similar to the monomial basis for a polynomial. The trigonometric polynomials are used in trigonometric interpolation to interpolate periodic functions. They are used in the discrete Fourier transform which is a special kind of trigonometric interpolation.
## Definition Let *a*_{n} be in **C**, 0 ≤ *n* ≤ *N* and *a*_{N} ≠ 0 then is called **complex trigonometric polynomial** of degree *N*. Using Euler's formula the polynomial can be rewritten as Analogously let *a*_{n}, *b*_{n} be in **R**, 0 ≤ *n* ≤ *N* and *a*_{N} ≠ 0 or *b*_{N} ≠ 0 then is called **real trigonometric polynomial** of degree *N*.
## Notes Using the relation *T*_{2N}(*x*) = *e*^{iNx}*t*_{N}(*x*) we can construct a bijective mapping between the *complex trigonometric polynomials* and the *real trigonometric polynomials*. Thus a trigonometric polynomial can be considered a periodic function on the real line, with period some multiple of 2π, or as a function on the unit circle. A trigonometric polynomial of degree *N* has a maximum of *N* roots in any open interval [*a*, *a* + 2π) with a in **R**. A basic result is that the trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm. This is a special case, for example, of the Stone-Weierstrass theorem. |