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Showing a given set of vectors forms a Parseval frame

  1. Feb 6, 2013 #1
    1. The problem statement, all variables and given/known data

    Show that the vectors
    ##\sqrt{\frac{2}{3}}(1,0), \sqrt{\frac{2}{3}}\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right), \sqrt{\frac{2}{3}}\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)##
    form a Parseval frame of ##\mathbb{R}^2##, but are neither linearly independent nor orthonormal

    2. Relevant equations

    The definition of Parseval frame, according to class notes, is
    "A sequence of vectors ##\displaystyle\left\{x_i \right\}_{i=1}^{k}## of an inner product space ##V## of dimension ##n (n\leq k)## is called a Parseval frame for ##V## if ##\forall x\in V##,
    ##||x||^2=\displaystyle\sum_{i=1}^{k}|\langle x,x_i \rangle|^2##.​

    3. The attempt at a solution

    I'm not quite sure how to interpret the definition. Or maybe I do, I just don't know how to implement it.

    What I've got so far:
    ##||x||^2 = \langle x,x\rangle = \displaystyle\sum_{i=1}^{3}|\langle x,x_i \rangle|^2##
    ##||x||^2 = \left|\left\langle x,\sqrt{\frac{2}{3}}(1,0)\right\rangle\right|^2 + \left|\left\langle x,\sqrt{\frac{2}{3}}\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right) \right\rangle\right|^2 + \left|\left\langle x,\sqrt{\frac{2}{3}}\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right) \right\rangle\right|^2##

    ##||x||^2 = \frac{2}{3} \left[\left|\left\langle x,(1,0)\right\rangle\right|^2 + \left|\left\langle x,\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right) \right\rangle\right|^2 + \left|\left\langle x,\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right) \right\rangle\right|^2\right]##

    Then, I suppose I take the dot product since ##\mathbb{R}^2## is my inner product space, so I let ##x = (x_1,x_2)##, where ##x_1,x_2\in\mathbb{R}##. Then I write
    ##||x||^2 = \frac{2}{3} \left[|x_1|^2 + \left| -\frac{1}{2}x_1 + \frac{\sqrt{3}}{2}x_2 \right|^2 + \left| -\frac{1}{2}x_1 - \frac{\sqrt{3}}{2}x_2 \right|^2\right]##

    Factoring out some constants gives me
    ##||x||^2 = \frac{2}{3} \left[|x_1|^2 + \frac{1}{4}\left|x_1 - \sqrt{3}x_2 \right|^2 + \frac{1}{4}\left|x_1 + \sqrt{3}x_2 \right|^2\right]##

    And that's all I've done. I'm not sure if I'm even doing this right. I've already shown that the vectors aren't linearly independent and orthonormal. Any ideas? Thanks.
     
  2. jcsd
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