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Question

Let $\hat f$ denote the Fourier transform of $f$ (non-unitary, angular frequency). I'm interested in a class of $\hat f$ that is characterized by the differential equation $$\hat f ' (t) = - i \alpha e^{-\beta t^2} \hat f(t) , \quad \alpha,\beta >0.$$

In particular, I am interested in finding the class of $\hat f$ that satisfies the differential equation and further satisfies the following properties on $f$:

  • $f(x)\in (0,1)$ for all $x \in \mathbb{R}$
  • $f'(x)<0$ for all $x \in \mathbb{R}$
  • $\lim_{x\to-\infty} f(x) = 1$ and $\lim_{x\to\infty} f(x) = 0$

Unfortunately, I don't know how (or if) I can translate those properties to $\hat f$. For instance, from the properties I can rule out the trivial solution to the differential equation $\hat f = 0$. But I don't know how to translate them in a more structured way.

More generally, the question is, how can I use the properties to refine the possible solutions to the differential equation? E.g., can I pin down a (possibly unique?) starting value for $\hat f (t)$ for some $t$?

Any help is much appreciated!


A first attempt

Suppose $\hat f(t) = c\cdot e^{h(t)}$ and substitute into the differential equation to conclude that $$ h(t) = \int -i \alpha e^{-\beta t^2} d t . $$ For any solution $c$, we can now compute the inverse Fourier transform, $$ f(x) = \frac{c}{2 \pi} g(x)$$ with $$g(x) = \int_{-\infty}^\infty e^{h(t) + i t x} d t, $$ and check whether it satisfies the required properties on $f$.

For now I haven't made much further progress. But it seems to me that, if any solution satisfies the properties, it has to be exactly one. Specifically, as the limit for $x\to -\infty$ requires $c = 2\pi / g(-\infty)$, we either have that for such $c$ all the other properties hold or not.

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