Trigonometric polynomial

Any function T of the form T(x) = a_0 + \sum_{n=1}^N a_n \cos nx + \sum_{n=1}^N b_n \sin nx \qquad (x \in \mathbb{R}) with complex-valued coefficients a_n and b_n and at least one of the highest-degree coefficients a_N and b_N non-zero, is called a complex trigonometric polynomial of degree N. The cosine and sine are the even and odd parts of the exponential of an imaginary variable, \cos nx = \tfrac12\bigl(e^{inx} + e^{-inx}\bigr), \quad \sin nx = -\tfrac12i\bigl(e^{inx} - e^{-inx}\bigr), so the trigonometric polynomial can alternately be written as T(x) = \sum_{n=-N}^N c_n e^{inx} \qquad (x \in \mathbb{R}), with complex coefficients and \quad c_k = \tfrac12(a_k - b_ki), \quad c_{-k} = \tfrac12(a_k + b_ki), for all from 1 to . If the coefficients a_n and b_n are real for all , then is called a real trigonometric polynomial. When using the exponential form, the complex coefficients satisfy c_{-n} = \overline{c}_{n} for all n\in[-N,N]. ==Properties==

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; this is a special case of the Stone–Weierstrass theorem. More concretely, for every continuous function and every there exists a trigonometric polynomial such that |f(z) - T(z)| for all . Fejér's theorem states that the arithmetic means of the partial sums of the Fourier series of converge uniformly to provided is continuous on the circle; these partial sums can be used to approximate . A trigonometric polynomial of degree has a maximum of roots in a real interval unless it is the zero function. == Fejér-Riesz theorem ==

The Fejér-Riesz theorem states that every positive real trigonometric polynomial t(x) = \sum_{n=-N}^{N} c_n e^{i n x}, satisfying t(x)>0 for all x\in\mathbb{R}, can be represented as the square of the modulus of another (usually complex) trigonometric polynomial q(x) such that: t(x) = |q(x)|^2 = q(x)\overline{q(x)}. Or, equivalently, every Laurent polynomial w(z)=\sum_{n=-N}^{N} w_{n}z^{n}, with w_n \in\mathbb{C} that satisfies w(\zeta)\geq 0 for all \zeta \in \mathbb{T} can be written as: w(\zeta)=|p(\zeta)|^2=p(\zeta)\overline{p(\zeta)}, for some polynomial p(z) = p_{0} + p_{1}z + \cdots + p_{N}z^{N}, and p(z) can be chosen to have no zeroes in the open unit disk \mathbb{D}. The Fejér-Riesz theorem arises naturally in spectral theory and the polynomial factorization w(\zeta)= p(\zeta)\overline{p(\zeta)} is also called the spectral factorization (or Wiener-Hopf factorization) of w(\zeta). ==Notes==