Let $a_1,a_2 \ldots a_n$ be a real-valued sequense with $a_i=O(N^{-1})$. How do I estimate Discrete Fourier Transform (DFT) of this sequence?

$$\hat{a}_{j}=\sum_{r=1}^{K}a_{r}\exp\Bigl(-2\pi ij\frac{r}{K}\Bigr),\qquad K=O(N^\alpha),\quad \alpha<1$$

Can I say that DFT sequence $\operatorname{Re}[\hat{a}_{i}]=O(N^{-1})$?