On the proof of Fejér-Riesz theorem

Question

I'm having a course about Analytic Number Theory, and I'm having trouble understanding the proof of Fejér-Riesz Theorem: http://people.virginia.edu/~jlr5m/Papers/FejerRiesz.pdf

First of all, I didn't understand why that polynomial $q(z)$ was used and why the observation that $q(0) \neq 0$ was relevant... And why the roots of modulus 1 have even multiplicity??

Can someone help? :)

Thanks!

Answer 1

Do you still require some input? If so, I can look at it and get back asap.

Hi. First let's note that the assumption implies that $\overline{c_{-n}}=c_n$ for all $n$ (start with $n=0$ and move up by differentiating, setting $t=0$ etc.) This then implies that $w(z)=\overline{w(1/\bar{z}})$. Thus if $z$ is a root, then $\frac{1}{\bar{z}}$ is also a root and so each must have the same multiplicity. When $z$ is on the unit circle then $\frac{1}{\bar{z}}$=$z$ so that root must appear at least twice or in "pairs" (which gives the even multiplicity of such roots). Note that $q$ has even degree and $0$ is never a root, so this makes things really nice. (The $q(0)\neq 0$ condition (equivalent to $c_n = \overline{c_{-n}}\neq 0$) is there to make the polynomial $w(z)$ to be of degree exactly $n$.)