Trigonometric polynomials
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 eix.
Any function T of the form
with an, bn in C for 0 ≤ n ≤ N, is called a complex trigonometric polynomial of degree N (Rudin 1987, p. 88). Using Euler's formula the polynomial can be rewritten as
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 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.
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