Ytukyg

I already asked a very similar question but I did not get any satisfying answer. Now I will ask the question a little different and explain my problem a little bit.

We have given $f:mathbb{R}^d rightarrow mathbb{R}$ with $f(x) = e^{-frac{1}{2}mid xmid^2}$.

Show that the fourier trafsform $hat f $ is given with $(sqrt{2 pi})^d f$.

So we actually dont know anything about the fourier transformation. We had a task last week which introduces the fourier tranform of a function but we wont talk about this topic in class. Its just an excursion.

We defined the fourier transform of $f in L^1(mathbb{R^d,mathbb{C}})$ as

$hat f(xi) := displaystyleint_{mathbb{R}^d}f(x)e^{-ixi cdot x} mu(dx)$

also we know we following:

$partial_jhat f(xi) = -idisplaystyleint_mathbb{R^d} x_j f(x)e^{-ixi cdot x} mu(dx)$ if $(x_1,...,x_d) mapsto x_j f(x)$ is integrable for every $j in lbrace 1,...,d rbrace$.

Thats everything we know about the fourier transform. There is no other thing we know. We even didnt know whats the purpose of this transformation. Its just a training task for multidimensional integration.

So back to the task. What I shell do is to take a look at $d = 1$ first by constructing a ordinary differential equation for $hat f$ which provides the solution $hat f = sqrt{2 pi} f$.

I literally got stuck there. The last time I asked this question with less detail one of the answers solved the integral for $hat f$ but using a method from complex analysis which Im not allowed to use.