Fourier Transform of Kernel Density Estimation - Convolution Theorem?
It's a bit contrived, but observe that by the "sifting property" of Dirac-delta function:

$$
K(x-X_j) = K(x) * \delta(x-X_j)
$$

where $*$ denotes convolution and $\delta$ is the Dirac-delta function.

Now you can apply Fourier convolution theorem and write:

$$
\mathcal{F}\{K(x-X_j)\}(t) = \kappa(t) e^{j t X_j}
$$

where $\mathcal{F}\{g(x)\}(t)$ denotes the Fourier transform of a function $g(x)$.

Finally using the linearity property of the Fourier transform:

$$
\mathcal{F}\{\hat{f}(x)\}(t) = \frac{1}{N} \sum_{j=1}^N \mathcal{F}\{K(x-X_j)\}(t) = \kappa(t) \sum_{j=1}^N e^{j t X_j}.
$$