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Math Help - Prof in functional analysis - help needed

  1. #1
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    Jun 2008
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    Prof in functional analysis - help needed

    Any help with this prof is most wellcome.

    Prove that, for any continuosly differentiable function f on [-phi, phi],

    │ ∫[-phi, phi] f(t)cos(t) f(t) sin (t) dt│
    ≤ (2*phi)^ * ({∫│f(t)^2 +│f(t)│^2})^

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  2. #2
    Member
    Joined
    Jan 2006
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    Gdansk, Poland
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    117
    I think I have the solution. I'm going to use Hoelder inequality
            \int_S \bigl| f(x)g(x)\bigr| \,\mathrm{d}x \le\biggl(\int_S |f(x)|^p\,\mathrm{d}x \biggr)^{\!1/p\;} \biggl(\int_S |g(x)|^q\,\mathrm{d}x\biggr)^{\!1/q}

    Where \frac{1}{p} + \frac{1}{q} = 1

    We see that
    f(t)\cos t - f'(t)\sin t = \mathrm{Re}\bigl[(f(t) + if'(t))(\cos t + i \sin t)\bigr]


    Hence
    I = \left|\int_{-\phi}^{\phi}\bigl(f(t)\cos t - f'(t)\sin t\bigr) dt \right|=  \left|\mathrm{Re} \left[ \int_{-\phi}^{\phi}\bigl(f(t) + if'(t)\bigr) e^{it}dt \right] \right| \leq

    \leq \left| \int_{-\phi}^{\phi}\bigl(f(t) + if'(t)\bigr) e^{it}dt \right|  \leq  \int_{-\phi}^{\phi}\bigl|f(t) + if'(t)\bigr|dt

    By Hoelder inequality we get
    I  \leq  \int_{-\phi}^{\phi}\bigl|f(t) + if'(t)\bigr|dt \leq \sqrt{\int_{-\phi}^{\phi}dt} \sqrt{\int_{-\phi}^{\phi}\bigl|f(t) + if'(t)\bigr|^2dt}

    Which, you can easily see implies the inequality we were to prove
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