trigonometric polynomial

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

T_N(x) = \sum_{n=0}^N a_n \cos (nx) + \mathrm{i}\sum_{n=0}^N a_n \sin(nx) \qquad (x \in \mathbf{R})

is called complex trigonometric polynomial of degree N. Using Euler's formula the polynomial can be rewritten as

T_N(x) = \sum_{n=0}^N a_n 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_N(x) = \frac{a_0}{2} + \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.

Notes

Using the relation

T_{2N}(x) = e^{\mathrm{i}Nx} 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.

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