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Math Help - Norm of powers of the Volterra Operator

  1. #1
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    Norm of powers of the Volterra Operator

    Hey there,

    For a course in Functional Analysis we were asked to show that the norm of the n-th power of the Volterra operator is less than or equal to 1 over n!.

    $V: C[0,1] \rightarrow C[0,1]$\\(Vf)(s) = \displaystyle{\int_0^s f(t)dt}
    Show $\left|\left|V^n\right|\right| \leq \frac{1}{n!}$

    Now I know how the operator norm is defined
    $\left|\left|T\right|\right| = \displaystyle{\sup_{\left|\left|x\right|\right|=1}  } \left|\left|Tx\right|\right|
    and I've been throwing it at this problem for at least 3 hours, but I can't seem to crack it.

    I've so far assumed that induction would be the path to choose but I'm starting to doubt that, yet I see no other method.

    I hope someone is able to nudge me in the right direction.

    Thanks,
    Insensus
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Norm of powers of the Volterra Operator

    Quote Originally Posted by Insensus View Post
    Hey there,

    For a course in Functional Analysis we were asked to show that the norm of the n-th power of the Volterra operator is less than or equal to 1 over n!.

    $V: C[0,1] \rightarrow C[0,1]$\\(Vf)(s) = \displaystyle{\int_0^s f(t)dt}
    Show $\left|\left|V^n\right|\right| \leq \frac{1}{n!}$

    Now I know how the operator norm is defined
    $\left|\left|T\right|\right| = \displaystyle{\sup_{\left|\left|x\right|\right|=1}  } \left|\left|Tx\right|\right|
    and I've been throwing it at this problem for at least 3 hours, but I can't seem to crack it.

    I've so far assumed that induction would be the path to choose but I'm starting to doubt that, yet I see no other method.

    I hope someone is able to nudge me in the right direction.

    Thanks,
    Insensus
    Maybe this will help you:

    SpringerLink - Integral Equations and Operator Theory, Volume 27, Number 4

    Hope that is free...
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  3. #3
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    México
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    Re: Norm of powers of the Volterra Operator

    How about this:

    Consider f_n(t)=t^n for n\geq 0 then it's not difficult to see that  V^k(f_n)(t) = \frac{t^{n+k}}{(n+k)...(n+1)}t^n from which we immediately deduce that \| V^k(p)(t)\| \leq \frac{1}{k!} where p is any polynomial. Now use Weierstrass' approximation theorem.
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  4. #4
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    Re: Norm of powers of the Volterra Operator

    That's very nice, thanks.
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