http://openproblemgarden.org/op/inequality_for_square_summable_complex_series Comments for "Inequality for square summable complex series " en http://openproblemgarden.org/op/inequality_for_square_summable_complex_series#comment-76066 <p>Where shall I send the £10 prize ?</p> Wed, 29 Oct 2014 20:49:40 +0100 tigris35711 comment 76066 at http://openproblemgarden.org http://openproblemgarden.org/op/inequality_for_square_summable_complex_series#comment-73221 <p>It's a simple application of the Shwartz inequality:</p> <p><img class="teximage" src="/files/tex/53db0acbd0ed85366a4842a1bd6f9f0c8c4bd039.png" alt="$$\sum_{k}\left|\sum_{l} \frac{1}{l+1}a_{2^k(2l+1)}\right|^2 \le $$" /> <img class="teximage" src="/files/tex/ed2567a94d0e6accd04b0139766fa82218eb60c5.png" alt="$$ \le \sum_{k}\left|\sum_{l} \frac{1}{l+1}\left|a_{2^k(2l+1)}\right|\right|^2 \le$$" /> Shwartz: <img class="teximage" src="/files/tex/52eb42db044e645552b4c8d6d0d2d3f3b2cb7d30.png" alt="$$ \le \sum_{k} \left(\sum_{l}\frac{1}{(l+1)^2}\right)\left(\sum_{h}|a_{2^k(2l+1)}|^2\right) = $$" /> <img class="teximage" src="/files/tex/ce011f08315775f2f7012f44b6e23d6d0a1f39e6.png" alt="$$ = \sum_{k} \frac{\pi^2}{6}\sum_{h}|a_{2^k(2l+1)}|^2 = $$" /> <img class="teximage" src="/files/tex/7a97e475239f7e22e37b3c87bc1375a9d15ef6a0.png" alt="$$ = \frac{\pi^2}{6} \sum_{k}\sum_{h}|a_{2^k(2l+1)}|^2 = $$" /> <img class="teximage" src="/files/tex/06adf98c7d9bfa2b45d9b55651a801eb66a54296.png" alt="$$ = \frac{\pi^2}{6} \sum_{n}|a_n|^2$$" /> because <img class="teximage" src="/files/tex/ed3c4678e1f6dbc3582145443caabc268ddffbfa.png" alt="$ A_k:=\{ 2^k(2l+1)| l\in \mathbb N\} $" /> is a partition of <img class="teximage" src="/files/tex/6b0b7a5cb47d1b439654473afd545ed2c9a05028.png" alt="$ \mathbb N^+ $" />. </p> Sun, 03 Nov 2013 15:31:54 +0100 Anonymous comment 73221 at http://openproblemgarden.org http://openproblemgarden.org/op/inequality_for_square_summable_complex_series <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/retkes_zoltan">Retkes</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/analysis">Analysis</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; For all <img class="teximage" src="/files/tex/98a062acfe8019375e85d2955372268be27cba3e.png" alt="$ \alpha=(\alpha_1,\alpha_2,\ldots)\in l_2(\cal{C}) $" /> the following inequality holds <img class="teximage" src="/files/tex/259cde58c5253d983f6a5ed781f307fe03260b4f.png" alt="$$\sum_{n\geq 1}|\alpha_n|^2\geq \frac{6}{\pi^2}\sum_{k\geq0}\bigg| \sum_{l\geq0}\frac{1}{l+1}\alpha_{2^k(2l+1)}\bigg|^2 $$" /></p> </div> </tr></td> </table> </td> </tr> </table> Retkes, Zoltan Inequality Analysis http://openproblemgarden.org/op/inequality_for_square_summable_complex_series#comment Tue, 25 Dec 2012 21:33:36 +0100 tigris35711 41335 at http://openproblemgarden.org