Let $\hat f$ be the Fourier transform of some function $f$. I am attempting to find the inverse Fourier transform of the following function: $$\hat f(x_1+ht,x_2+ht,\dots,x_n+ht)$$ with respect to $h$. Another way to write it may be $\hat f(\boldsymbol{y})$ where $y_i=x_i+ht$. My first thought is to figure out what the answer is in just one dimension, i.e. for $\hat f(x+ht)$, and then work from there. I was able to successfully solve it for the 1D case with the following work.
My work for 1-D case:
Define the Fourier transform to be $$\hat g(h)=\mathcal{F}_\eta\left\{g(\eta)\right\}(h)=\int_{-\infty}^\infty g(\eta)\,e^{-2\pi ih\eta}d\eta\;.$$ Let $p=x+ht$, so now the inverse Fourier transform of $\hat f$ is $$\begin{align}\mathcal{F}^{-1}_h\left\{\hat f(x+ht)\right\}(\eta)&=\frac{1}{t}\int_{-\infty}^\infty \hat f(p)\,e^{2\pi i\eta \left(\frac{p-x}{t}\right)}dp\\&=\frac{1}{t}\,e^{-2\pi ix\eta/t}\int_{-\infty}^\infty\hat f(p)\,e^{2\pi i\eta p/t}dp\\&=\frac{1}{t}\,e^{-2\pi ix\eta/t}f\left(\frac{\eta}{t}\right)\end{align}$$ This result seems useful and is really just an exercise in the stretch and shift theorems. Though, I run into problems in the $n$-dimensional case.
My attempt for $n$-D case:
Use the vector $\boldsymbol{y}$ as defined above. Then $h=\frac{y_i-x_i}{t}$. Already there is a problem. This would imply $dh=\frac{1}{t}dy_i$, meaning the integration is only with respect to the coordinate I choose. $$\mathcal{F}^{-1}\left\{\hat f\right\}=\frac{1}{t}\,e^{-2\pi ix_i\eta/t}\int_{-\infty}^\infty\hat f(\boldsymbol{y})\,e^{2\pi i\eta y_i/t}dy_i$$ Not sure if this means anything. I also don't know how to evaluate it. So then I attempted to have the integral representative of all the coordinates. To this end, I rewrote $\boldsymbol{y}$ as $\boldsymbol{y}=\boldsymbol{x}+ht\boldsymbol{1}$, where $\boldsymbol{1}$ is a vector of ones. Then by multiplying through by $\boldsymbol{1}$, we can get to the following representation for the inverse Fourier transform: $$\mathcal{F}^{-1}\left\{\hat f\right\}=\frac{1}{nt}\,e^{-2\pi i\eta\boldsymbol{x}\cdot\boldsymbol{1}/(nt)}\sum_{i=1}^n\int_{-\infty}^\infty\hat f(\boldsymbol{y})\,e^{2\pi i\eta \boldsymbol{y}\cdot\boldsymbol{1}/(nt)}dy_i$$ Now it is just a sum of the integrals previously which I have no idea what to do with. Help would be much appreciated for this problem. Keep in mind the transform is done with respect to $h$, a 1-D variable which is why I was avoiding an $n$-D transform when attempting this.