If we square a function, what happens to its Fourier transform?

Question

Suppose we have a function $f$ and its Fourier transform: $$F[f](t) = \displaystyle\int_{-\infty}^\infty f(x)\cdot e^{-itx}dx$$ Now we are interested in: $$F[f^2](t) = \displaystyle\int_{-\infty}^\infty f^2(x)\cdot e^{-itx}dx$$ Can we express $F[f^2](t)$ in terms of $F[f](t)$? Or if that isn't possible, can we approximate $F[f^2](t)$ using $F[f](t)$?

The only property of the Fourier-transform I could find that might be helpful is Parseval's equation:

$$\int_{-\infty}^\infty f(t)^2 dx = \frac{1}{2 \pi} \int_{-\infty}^\infty [Ff(t)]^2 dx$$

I was hoping to use this to write something like: $$\int_{-\infty}^\infty F[f^2](t) \stackrel{?}{=}\int_{-\infty}^\infty f(t)^2 dx = \frac{1}{2 \pi} \int_{-\infty}^\infty (F[f](t))^2 dx$$ Then maybe use this in an approximation.

Could it really be as simple as: $Ff^2 = ( Ff )^2$? Just the component-wise squaring? — dimpol, May 09 '17 at 14:44
No, that isn't correct. I've posted the form for the Fourier Transform of $f^2$. — Mark Viola, May 09 '17 at 14:49

score 12 · Accepted Answer · answered May 09 '17 at 14:48

12

From the convolution theorem, we can write

$$\int_{-\infty}^\infty f^2(x)e^{-itx}\,dx=\frac{1}{2\pi}\int_{-\infty}^\infty F(t')F(t-t')dt'$$

The Theorem is actually more general than this. If $f$ and $g$ have Fourier Transforms $F$ and $G$, respectively, then the Fourier Transform of the product $fg$ is given by

$$\int_{-\infty}^\infty f(x)g(x)e^{-itx}\,dx=\frac{1}{2\pi}\int_{-\infty}^\infty F(t')G(t-t')dt'$$

answered May 09 '17 at 14:48

Mark Viola

179,405

What about if $f(x) =e^{-\alpha |x|^2}$? How to calculate explicity $\mathcal{F}(f^2)(\xi)$? – Lucas Linhares Nov 27 '22 at 10:47
@Thiago The square of $f$ is $e^{-2\alpha |x^2|}$ – Mark Viola Nov 27 '22 at 16:08
My bad. I meant: If $f(x) = e^{-\alpha |x|^2.|x|^2}$, is there any rule to calculate $\hat{f}$ ? – Lucas Linhares Nov 27 '22 at 19:58
Not in terms of elementary functions. – Mark Viola Nov 27 '22 at 21:54
Even in terms on non elementary functions, do you indicate a good reference with this things? – Lucas Linhares Nov 27 '22 at 23:22
1

@Thiago This is not relevant to the question asked here. If you include the generalized hypergeometric function, then yes. – Mark Viola Nov 28 '22 at 02:23

If we square a function, what happens to its Fourier transform?

1 Answers1