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

  1. Nov 27, 2014 #1
    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)##.
     
  2. jcsd
  3. Nov 28, 2014 #2

    WWGD

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    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.
     
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