Consider a real-valued function $h_k(t)$ defined by $$ t \mapsto \exp\left(\,{t^{2} \over 2}\right)\, \frac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}e^{-t^2}\quad \mbox{where}\quad k \in \mathbb{N}_0. $$ I need to show that $$\mathcal{F}h_k = (-i)^k\sqrt{2\pi}h_k$$ where $\mathcal{F}$ is a Fourier transform, i.e $\mathcal{F}: L^1(\mathbb{R}) \to L^{\infty}(\mathbb{R})$ is defined by $f(x) \mapsto \hat{f}(\xi) = \int_{\mathbb{R}}f(x)e^{-ix\xi}\ dx$. First, as a hint, I need to show that $h_{k+1}(t) = h^{\prime}_k(t) -th_k(t)$ for all $t \in \mathbb{R}$. I manage to show this hint, but the problem is, how is this helpful. There is $h_{k+1}$ in the equation, but I only want $\mathcal{F}h_k$. The operator $\mathcal{F}$ is linear, so $$\mathcal{F}h_{k+1} = \mathcal{F}h^{\prime}_k - \mathcal{F}(th_k).$$ I know that $\mathcal{F}h^{\prime}_k(x) = \widehat{h^{\prime}_k}(\xi) = i\xi\widehat{h_k}(\xi)$ and $-\mathcal{F}(th_k) = \widehat{-th_k}(\xi) = -i(\widehat{h_k})^{\prime}(\xi)$. So, $$\widehat{h_{k+1}}(\xi) = i\xi\widehat{h_k}(\xi)-i(\widehat{h_k})^{\prime}(\xi).$$ I have no idea what to do next. If anyone has an idea, could you please share your idea on how to solve this problem?
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Shouldn't your definition be $$h_n(t) = e^{t^2/2} \frac{d^n}{dt^n}e^{-t^2}$$ – Cameron L. Williams Nov 17 '20 at 18:55
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$e^{-t^2/2}$ should indeed be $e^{t^2/2}$. See this answer. – Maxim Nov 17 '20 at 19:05
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I have edited the question. It should be indeed $e^{t^2/2}$ – Vicky Nov 18 '20 at 09:46
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First of all see how it works with the generating function of the Hermite functions
Your exercice is saying that if $\hat{h} = ch$ then $\widehat{ h'-th} = it \hat{h}-i \hat{h}' = it c h- ich'$
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