1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ You should look up $n$ dimensional Fourier transform. You are only calculating the transform with respect to a single coordinate, not $\mathbb{R}^n$ as I expect you intended. I would provide further assistance, but it is unclear if your issue is with basic multivariable calculus or with Fourier series, because it should be clear that you aren't defining the transform correctly. $\endgroup$ – adfriedman Aug 29 '17 at 1:41
  • $\begingroup$ @adfriedman Thanks for the recommendation. I have tried reading up on $n$-D Fourier transforms and will continue my reading. Though $h$ is a single variable, not a vector, which is why I think the transform can only be for a single coordinate. Is this false thinking? $\endgroup$ – MasterYoda Aug 29 '17 at 1:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.