General Problems on Trigonometric Polynomials

A cosine polynomial of order $n$ may always be written in the form $P(\cos \vartheta )$, where $P(x)$ is a polynomial of degree $n$. The converse is also true.

A sine polynomial of order $n$ may always be written in the form $\sin \vartheta \, P(\cos \vartheta )$, where $P(x)$ is a polynomial of degree $n-1$. The converse is also true.

The product of two trigonometric polynomials of order $m$ and $n$ respectively is a trigonometric polynomial of order $m+n$.

A trigonometric polynomial of order $n$ with real coefficients $\lambda_r, \mu_s$, $r = 0, 1, 2, \ldots$, $s = 1, 2, \ldots$,

may be written in the form

where $G(z) = u_0 + u_1 z + u_2 z^2 + \cdots + u_{2n} z^{2n}$ denotes a polynomial of degree $2n$ which remains unchanged if one forms the reciprocal polynomial and simultaneously replaces the coefficients by their complex conjugates:

Calculate the coefficients $u_0, u_1, u_2, \ldots , u_{2n}$.

Denote by $G(z)$ a polynomial of degree $2n$ satisfying the identity

Then for $\vartheta$ real

is a trigonometric polynomial in $\vartheta$ of order $n$ with real coefficients.

Let $G(z)$ be a polynomial of degree $2n$ and

How are the zeros of $G(z)$ distributed in the complex plane?

A trigonometric polynomial of order $n$ with real coefficients

in which $\lambda_n$ and $\mu_n$ are not both zero, has exactly $2n$ zeros, if both real and complex values are admitted and multiple zeros are counted according to their multiplicity.

Determine all trigonometric polynomials of order $n$

which have real coefficients and satisfy identically in $\alpha$ and $\beta$ the relation