Why is this expression real valued and nonnegative?

Question

$$ g(s)= \begin{cases} 1.4, &\text{for } s=0\\ (0.9)^{|s|},&\text{for } 1\leq |s|\leq 9\\ 0, &\text{else.} \end{cases} $$ I want to show that $$ \sum_{s\in\mathbb{Z}}e^{-i\lambda s}g(s)\geq 0 $$ for all $\lambda\in [-\pi,\pi]$, implying that the expression is real valued.

Answer 1

You have \begin{align} \sum_{s\in\mathbb Z} e^{-i\lambda s}g(s) &= \sum_{s=-9}^{-1} e^{-i\lambda s}(0.9)^{-s} + 1.4 + \sum_{s=1}^{9} e^{-i\lambda s}(0.9)^{s} = \\ &= 1.4 + \sum_{s=1}^{9} \big(e^{-i\lambda s} + e^{i\lambda s}\big)(0.9)^{s} = \\ &= 1.4 +2 \,{\rm Re} \Big(\sum_{s=1}^{9} e^{i\lambda s}(0.9)^{s}\Big) \end{align} From this expression it's already visible that the sum is real-valued.

We have \begin{align} \sum_{s\in\mathbb Z} e^{-i\lambda s}g(s) &= 1.4 +2 \,{\rm Re} \Big(\sum_{s=1}^{9} (0.9e^{i\lambda})^s\Big) = \\ &= 1.4 + 2 \,{\rm Re} \Big(\frac{0.9e^{i\lambda} - (0.9e^{i\lambda})^{10}}{1- 0.9e^{i\lambda}}\Big) \end{align} It turns out to be positive for most $\lambda$, but not for all. For example for $\lambda =\frac{\pi}{6}$ we have \begin{align} \sum_{s\in\mathbb Z} e^{-i\pi s/6}g(s) &= 1.4 + 2 \,{\rm Re} \Big(\frac{0.9e^{i\pi/6} - (0.9)^{10}e^{10i\pi/6}}{1- 0.9e^{i\pi/6}}\Big) = \\ &\approx 1.4 + 2 \,{\rm Re} \Big(-0.815898 + 1.74456 i\Big) = \\ &= -0.231797 \end{align}