Trigonometric polynomials Polynomial

a trigonometric polynomial finite linear combination of functions sin(nx) , cos(nx) n taking on values of 1 or more natural numbers. coefficients may taken real numbers, real-valued functions.

if sin(nx) , cos(nx) expanded in terms of sin(x) , cos(x), trigonometric polynomial becomes polynomial in 2 variables sin(x) , cos(x) (using list of trigonometric identities#multiple-angle formulae). conversely, every polynomial in sin(x) , cos(x) may converted, product-to-sum identities, linear combination of functions sin(nx) , cos(nx). equivalence explains why linear combinations called polynomials.

for complex coefficients, there no difference between such function , finite fourier series.

trigonometric polynomials used, example in trigonometric interpolation applied interpolation of periodic functions. used in discrete fourier transform.