Eigenvectors of Fourier transform operator #F:L^2\to L^2#

[DavideGenoa]

Hi, friends! In order to find an orthogonal basis of eigenvectors of the Fourier transform operator $F : L_2(\mathbb{R})\to L_2(\mathbb{R}),f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x$ for Euclidean separable space $L_2(\mathbb{R})$, so that $F$ would be represented by an infinite diagonal matrix, A.N. Kolmogorov and S.V. Fomin, in their Элементы теории функций и функционального анализа (Elements of the theory of functions and functional analysis), pp. 442-445 here, use functions that are, up to a constant factor, the Hermite functions, which I know to constitute an orthogonal basis of $L_2(\mathbb{R})$.

While looking for such an orthogonal basis of eigenvectors, Kolmogorov and Fomin search for Schwartz functions, belonging to $S_\infty\subset L_2(\mathbb{R})$, which is dense everywhere, in the form $w(x)e^{-x^2/2}$ where $w$ is a polynomial, satisfying the equation

$\frac{d^2f}{dx^2}-x^2f=\mu g\quad\quad\text{equation }(3)$
​

where $\mu$ is a constant, which may well be not the same for all solutions. Such an equation change into $\frac{d^2g}{d\lambda^2}-\lambda^2g=\mu f$ when acted upon by operator $F$, where I have written $g:= F(f)$. It is shown that the polynomials $w$ of such (Hermite up to a constant factor) functions satisfy equation (3) for $\mu=-(2n+1)$ when $\deg w=n$ (let us call such a polynomial $P_n$) and they have non-null coefficients $a_k$ of the variable $x^k$ only for the $k$'s of the same oddity of $n$. It had also previously proved (p. 401) that $P_n$ is, up to a constant, $(-1)^ne^{x^2}\frac{d^n e^{-x^2}}{dx^n}$ (I am writing this in the case it can be used to prove that $P_n(x)e^{-x^2/2}$ defines an eigenvector of $F$).

Kolmogorov-Fomin's says that the fact that the $P_ne^{-x^2/2}$ are eigenvectors of $F$ and their eigenvalues are $\pm\sqrt{2\pi}$, $\pm i\sqrt{2\pi}$ derives from the following fact:

1. Equation (3) is invariant with respect to transformation $F$.
2. Equation (3) has got, up to a constant factor, one solution of the form $P_ne^{-x^2/2}$.
3. The Fourier transform maps $x^ne^{-x^2/2}$ to $i^n\sqrt{2\pi}\frac{d^n}{dx^n}e^{-x^2/2}$ (as [I knew][3], by using the fact that $F[e^{-x^2/2}](\lambda)=\sqrt{2\pi}e^{-\lambda^2/2}$).

The proof of this derivation contained in the book only says that $F^4[P_n e^{-x^2/2}]=4\pi^2 P_n e^{-x^2/2}$, which I know to be true for all #f\in L_2(\mathbb{R})#, but I could not see how this is related to the fact that the eigenvalues are the fourth roots of $4\pi^2$. I have found some resources talking about the topic on line, but nothing using Kolmogorov-Fomin's argument, which I would like to understand... Does anybody understand and can explain it? I heartily thank you!

P.S.: If it were useful to understand what Kolmogorov-Fomin's says, I know that $\forall f\in S_\infty$, $k\in\mathbb{N}$ $F[f^{(k)}](\lambda)=(i\lambda)^kF[f](\lambda)$ and $\frac{d^k}{d\lambda^k}F[f](\lambda)=(-i)^kF[x^k f](\lambda)$.

 

[WWGD]

Davide, just a suggestion: make the question more concise and then maybe do a second post with additional content, or wait until someone asks a followup. This may improve your chances of getting an answer.