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
, i.e., Laurent polynomials in
under the change of variables
Any function T of the form
and at least one of the highest-degree coefficients
non-zero, is called a complex trigonometric polynomial of degree N.[1] Using Euler's formula the polynomial can be rewritten as
Analogously, letting coefficients
non-zero or, equivalently,
is called a real trigonometric polynomial of degree N.[2][3] A trigonometric polynomial can be considered a periodic function on the real line, with period some divisor of
, or as a function on the unit circle.
Trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm;[4] this is a special case of the Stone–Weierstrass theorem.
More concretely, for every continuous function
there exists a trigonometric polynomial
Fejér's theorem states that the arithmetic means of the partial sums of the Fourier series of
converge uniformly to
is continuous on the circle; these partial sums can be used to approximate
A trigonometric polynomial of degree
roots in a real interval
[5] The Fejér-Riesz theorem states that every positive real trigonometric polynomial
, can be represented as the square of the modulus of another (usually complex) trigonometric polynomial
Or, equivalently, every Laurent polynomial
w ( ζ ) ≥ 0
ζ ∈
w ( ζ ) =