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Fourier transform in 3d

  1. Feb 25, 2012 #1
    1. The problem statement, all variables and given/known data
    Show that [itex]|\tilde{I}(\vec{k})|^2\leq CP^2[/itex]
    Where C denotes a constant,
    Using this inequality: [itex]\int f(\vec{r})^*g(\vec{r})\,\text{d}\vec{r}\leq \int f^*f\text{d}\vec{r}\,\int gg^*\text{d}\vec{r}[/itex]
    Where k denotes the fourier transform from r->k(in 3d)
    R is assumed positive definite, real, symmetric and normalized like [itex]\int R(r)dr = 1[/itex].


    2. Relevant equations
    I already know that [itex] P = \int\limits_{-\infty}^{\infty} I\text{d}r[/itex]
    And that [itex] \tilde{N}(I) = \tilde{R}(\vec{k})\tilde{I}(\vec{k})[/itex]
    And that [itex] \int N(I)I\,\text{d}\vec{r} = (2\pi)^{3/2}\int \tilde(R(\vec{k})|\tilde{I}(\vec{k})|^2\text{d}\vec{k}[/itex]
    The Nonlocal Gross-Pitaveski equation:
    [itex] i\dfrac{\partial u(x,t)}{\partial t} +\nabla^2 u(x,t) -V(r)u + N(I)u = 0[/itex]
    The convolution theorem and Parsevals theorem is also relevant.

    3. The attempt at a solution
    I tried many attempts, if anyone just could give a hint, for example this was one of the wrongs:
    [itex] |\int \dfrac{d\tilde{I}}{d\vec{k}}\,\text{d}\vec{k}|^2 = |\tilde{I}(\vec{k})|^2\leq \int 1\times 1^*\text{d}\vec{k} \int \dfrac{d\tilde{I}^*}{d\vec{k}}\dfrac{d\tilde{I}}{d\vec{k}}[/itex]
    Or maybe with a delta function on it:
    [itex] |\int \tilde{I}\delta^3(\vec{y}-\vec{k})\text{d}\vec{k}|^2 = |\tilde{I}(\vec{k})|^2 \leq \int |I(k)|^2\text{d}\vec{k}[/itex]
    But i really can't see what i should start with under the absolute squared sign.
    Any ideas or better hints will be like gold for me.
     
    Last edited: Feb 25, 2012
  2. jcsd
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