I would like to simplify if possible the spatio-temporal triple correlation of the following function:
$$f(\vec{x},t)=\delta(\vec{x}-\vec{x}_0(t)) \otimes f_p(\vec{x})$$
where $\delta$ is the Dirac delta, $f_p$ is a generic function (i.e. a 2D Gaussian) and $\otimes$ represents the convolution operator.
I define the triple correlation as:
$$C(\vec{\xi}_1,\vec{\xi}_2,\tau_1,\tau_2):=<f(\vec{x},t)f(\vec{x}+\vec{\xi}_1,t+\tau_1)f(\vec{x}+\vec{\xi}_2,t+\tau_2)>_{\vec{x},t}$$
The spatial bispectrum of $C$ is
$$B_{\vec{x}}=<\hat{f}(\vec{k}_1,t)\hat{f}(\vec{k}_2,t+\tau_1)\hat{f}(-\vec{k}_1-\vec{k}_2,t+\tau_2)>_{t}$$
where
$$\hat{f}(\vec{k}_1,t)=\mathcal{F}\{f(\vec{x},t)\}=\mathcal{F}\{\delta(\vec{x}-\vec{x}_0(t)) \otimes f_p(\vec{x})\}=\mathcal{F}\{\delta(\vec{x}-\vec{x}_0(t))\}\mathcal{F}\{f_p(\vec{x})\}=\hat{f}_p(\vec{k}_1)\exp{(-2\pi i\vec{k}_1\cdot\vec{x}_0(t))}$$
Then it follows that:
$$B_{\vec{x}}=\hat{f}_p(\vec{k}_1)\hat{f}_p(\vec{k}_2)\hat{f}_p(-\vec{k}_1-\vec{k}_2)<\exp{(-2\pi i\vec{k}_1\cdot\vec{x}_0(t))}\exp{(-2\pi i\vec{k}_2\cdot\vec{x}_0(t+\tau_1))}\exp{(2\pi i(\vec{k}_1+\vec{k}_2)\cdot\vec{x}_0(t+\tau_2))}>_{t}$$
Is it ok until now? Could I do something more? Thanks