Hello im struggling to proove, that there exsists an constant $C$, so that for every $N \in \mathbb {N}$ and every $ x \in [0,N\pi]$, the following holds: $$\sum_{k=0}^N \frac {k^2}{N^3} \cos \left(\frac{k}{N}x\right) \leq C\cdot(x^{-1}+x^{-2}+x^{-3})$$
The Approximation is used in the following article: https://www.researchgate.net/publication/267676914_CLT_for_Crossings_of_random_trigonometric_Polynomials
Let us consider $f(y)=y^2\cos(xy)$. Then $$ \int_{0}^{1} f(y)\,dy = \frac{2x\cos(x)+(x^2-2)\sin(x)}{x^3}\ll 2\left(\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}\right) $$ and the difference between this integral and the associated Riemann sum $$ \frac{1}{N}\sum_{k=0}^{N}\left(\frac{k}{N}\right)^2 \cos\left(x\frac{k}{N}\right) $$ is bounded by $\frac{x}{N^2}$ due to the Lipschitz-continuity of $\cos$ and summation/integration by parts.
It follows that your bound holds as soon as $N\gg|x|$.