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Math Help - Approximation in Sobolev Spaces

  1. #1
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    Approximation in Sobolev Spaces

    I just want to know how it follows that v^{\epsilon} \in C^{\infty}(\bar{V})? I could see how v^{\epsilon} \in C^{\infty}(V) by using the translations, but I'm having difficulty seeing how it extends to \bar{V}, since it says that u_{\epsilon}(x) := u(x^{\epsilon}) \text{ for } x \text{ in } V, we are mollifying on V. Do you have any idea of how this follows? Thanks for the assistance.
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Approximation in Sobolev Spaces

    What the symbolism denotes is that the limit \lim_{x\rightarrow y \in \overline{V}}D^{\alpha}v^{\epsilon} necessarily exists for all \alpha (as D^{\alpha}u is defined on \overline{V}).
    Last edited by Rebesques; December 2nd 2013 at 12:28 PM.
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