Trigonometric Polynomials
Definition
A trigonometric polynomial
of order $n$ is a function $u: \R \to \R $
for which there exists $\alpha _0, \alpha _1,
\dots , \alpha _n, \beta _1, \beta _2, \dots ,
\beta _n \in \R $ so that
\[
u(x) = \alpha _0 + \sum_{k = 1}^{n} \parens*{\alpha _k
\cos(kx) + \beta _k \sin(kx)}.
\] \[
\begin{aligned}
u(x) = \alpha _0 +& \alpha _1\cos x + \beta _1\sin x +
\alpha _2 \cos2x + \beta _2 \sin2x + \\ &\cdot s + \alpha _n
\cos nx + \beta _n \sin nx.
\end{aligned}
\]