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Math Help - Differential Operator Norm

  1. #1
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    Differential Operator Norm

    Any Idea that help in solving this problem is Highly appreciated
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  2. #2
    Super Member girdav's Avatar
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    Re: Differential Operator Norm

    Consider a function \phi smooth, such that \int_{\mathbb R}\phi^2(t)dt=1, and put \phi_n(t)=\sqrt n\phi(nt). Then \lVert \phi_n\rVert_{L^2}=1, but \frac d{dt}\phi_n(t)=n^{\frac 32}\phi(nt) has a norm L^2 which cannot be bounded by a constant which doesn't depend on n.

    For the second problem, note that ||f||^2\geq ||\frac d{dt}f||^2_{L^2}.
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