Trigonometric polynomial.html

In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite 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

Any function T of the form

$T(x) = a_0 + \sum_{n=1}^N a_n \cos (nx) + \mathrm{i}\sum_{n=0}^N b_n \sin(nx) \qquad (x \in \mathbf{R})$

with a_n, b_n 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

$T(x) = \sum_{n=-N}^N c_n \mathrm{e}^{\mathrm{i}nx} \qquad (x \in \mathbf{R}).$

Analogously let a_n, b_n be in R, 0 ≤ n ≤ N and a_N ≠ 0 or b_N ≠ 0 then

$t(x) = a_0 + \sum_{n=1}^N a_n \cos (nx) + \sum_{n=1}^N b_n \sin(nx) \qquad (x \in \mathbf{R})$

is called real trigonometric polynomial of degree N (Powell 1981, p. 150).

Notes

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 basic result is that the trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm (Rudin 1987, Thm 4.25); this is a special case of the Stone–Weierstrass theorem. More concretely, for every continuous function ƒ and every ε > 0, there exists a trigonometric polynomial T such that |ƒ(z) − T(z)| < ε for all z. Fejér's theorem states that the arithmetic means of the partial sums of the Fourier series of ƒ converge uniformly to ƒ, thus giving an explicit way to find an approximating trigonometric polynomial T.

A trigonometric polynomial of degree N has a maximum of 2N roots in any open interval a, a + 2π) with a in R, unless it is the zero function (Powell 1981, p. 150).

References

Powell, Michael J. D. (1981), Approximation Theory and Methods, Cambridge University Press, ISBN 978-0-521-29514-7
Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, MR 924157, ISBN 978-0-07-054234-1 .