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Math Help - [SOLVED] B-Splines

  1. #1
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    [SOLVED] B-Splines

    Let B_i^k be the B-splines of degree k. Then prove
    \sup_{-\infty<x<\infty}\left|\sum_{i=-\infty}^{\infty}c_i B_i^k(x)\right| \le \sup_{-\infty<i<\infty} |c_i|.

    I applied the triangle inequality and got
    \sup_{-\infty<x<\infty}\left|\sum_{i=-\infty}^{\infty}c_i B_i^k(x)\right| \le \sup_{-\infty<x<\infty} \sum_{i=-\infty}^{\infty} |c_i|B_i^k(x).
    Beyond this point I am clueless.

    EDIT: It seems like I should be using
    \sum_{i=-\infty}^{\infty} B_i^k(x) = 1\quad \forall x\in\mathbb{R}
    Last edited by lvleph; March 2nd 2010 at 09:53 AM.
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  2. #2
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    Just in case someone looks at this and was wondering the solution.
    <br />
\sup_{-\infty<x<\infty}\sum_{i=-\infty}^{\infty}|c_i| B_i^k(x) \le \sup_{-\infty<i<\infty} |c_i| \cdot \sup_{-\infty<x<\infty} \sum_{i=-\infty}^{\infty} B_i^k(x).<br />
    Thus,
    <br />
\sup_{-\infty<x<\infty}\sum_{i=-\infty}^{\infty}|c_i| B_i^k(x) \le \sup_{-\infty<i<\infty} |c_i|<br />
    which is the desired result.
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