Location and number of zeros of trigonometric polynomials

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Problem 5.27 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 2 "Reversals of Sign of a Function"

The trigonometric polynomial

f(x) = a_0 + a_1 \cos x + a_2 \cos 2x + \cdots + a_n \cos nx

with real coefficients $a_0, a_1, a_2, \ldots , a_n$ has only real zeros if

\left|a_0\right| + \left|a_1\right| + \left|a_2\right| + \cdots + \left|a_{n-1}\right| < a_n.

Problem 6.13 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 6.2 "General Problems on Trigonometric Polynomials"

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

z^{2n} \overline{G}(z^{-1}) = G(z).

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

Problem 6.14 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 6.2 "General Problems on Trigonometric Polynomials"

A trigonometric polynomial of order $n$ with real coefficients

g(\vartheta ) = \lambda _0 + \lambda _1 \cos \vartheta + \mu _1 \sin \vartheta + \lambda _2 \cos 2\vartheta + \mu _2 \sin 2\vartheta + \cdots + \lambda _n \cos n\vartheta + \mu _n \sin n\vartheta ,

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.

Remark.

$\vartheta$ and $\vartheta + 2\pi$ are regarded as not distinct.

Problem 6.19 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 6.3 "Some Special Trigonometric Polynomials"

Where do the zeros of the following trigonometric polynomials lie:

(1)

$\dfrac {1}{2} + \cos \vartheta + \cos 2\vartheta + \cdots + \cos n\vartheta$ ;

(2)

$\cos \vartheta + \cos 2\vartheta + \cdots + \cos n\vartheta$ ;

(3)

$\cos \vartheta + \cos 3\vartheta + \cos 5\vartheta + \cdots + \cos (2n-1)\vartheta$ ;

(4)

$\sin \vartheta + \sin 2\vartheta + \cdots + \sin n\vartheta$ ;

(5)

$\sin \vartheta + \sin 3\vartheta + \cdots + \sin (2n-1)\vartheta$ ;

(6)

$\dfrac {n+1}{2} + n \cos \vartheta + (n-1) \cos 2\vartheta + \cdots + \cos n\vartheta$ ?

References

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.