I'm taking a signal processing class and we were studying windowing functions. One of the functions is the triangular function:
$$w(n) = \left\{\begin{array}{ll} \frac{2n-1}{L} & 1\le n\le L/2\\ 2-\frac{2n-1}{L} & L/2+1<n<L \end{array} \right.$$
where L is the number of points you want.
We were tasked to find L=64 points and then use the values obtained as coefficients of a polynomial and find the roots (with Matlab -- the professor isn't a sadist).
So, this polynomial will be palindromic.I found that there were no unique roots (I can post the values if needed). In fact for L=64, there are 31 roots. Investigating further, no matter what L I chose I got repeated roots.
This wasn't part of the assignment or anything but I'm just curious about it. My initial hunch would be because the polynomial is symmetric there would be repeated roots. The Binomial Theory is what came to mind. However I know that not all symmetric polynomials have repeated roots so I was curious how this can be explained.
I'm an engineer by background so I apologize if any of my nomenclature is incorrect or the question isn't worded properly.
EDIT:
So the coefficients of my polynomial are $w(n)$ where $n = 1,2,...,L$ i.e.
$w(1), w(2)....,w(L)$
$w(1)x^{L-1} + w(2)x^{L-2} +...+ w(L) = 0$