What square-summable sequences are "sinc-summable"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T01:27:05Z http://mathoverflow.net/feeds/question/79332 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79332/what-square-summable-sequences-are-sinc-summable What square-summable sequences are "sinc-summable"? Ricky Demer 2011-10-27T23:46:52Z 2011-10-27T23:46:52Z <p>$\operatorname{sinc} : \mathbb{R} \to \mathbb{R} \;\;$ is defined by <code>$\;\; \operatorname{sinc}(x) \; = \; \begin{cases} 1 &amp; \text{if }\:\;x=0 \\ \\ \frac{\operatorname{sin}(x)}x &amp; \text{else} \end{cases} \;\;$</code> . <br><br><br> For what square-summable sequences $\langle x_0,x_1,x_2,x_3,...\rangle$ of complex numbers is it the case that</p> <p>$\displaystyle\lim_{h\to 0^+} \left(\displaystyle\sum_{n=0}^{\infty} \: (\operatorname{sinc}(n\cdot h)\cdot x_n)\right) \;$ exists and is finite? <br><br><br> (In other words, when is it the case that the Lebesgue mean at zero of a member of $\; \operatorname{L}_2[-\pi,\pi] \;$ <br> exists and is finite, in terms of the coefficients of that member's exponential fourier series?)</p>