$\operatorname{sinc} : \mathbb{R} \to \mathbb{R} \;\;$ is defined by $\;\; \operatorname{sinc}(x) \; = \; \begin{cases} 1 & \text{if }\:\;x=0 \\ \\ \frac{\operatorname{sin}(x)}x & \text{else} \end{cases} \;\; $ .

For what square-summable sequences $\langle x_0,x_1,x_2,x_3,...\rangle$ of complex numbers is it the case that

$\displaystyle\lim_{h\to 0^+} \left(\displaystyle\sum_{n=0}^{\infty} \: (\operatorname{sinc}(n\cdot h)\cdot x_n)\right) \; $ exists and is finite?

(In other words, when is it the case that the Lebesgue mean at zero of a member of $\; \operatorname{L}_2[-\pi,\pi] \;$

exists and is finite, in terms of the coefficients of that member's exponential fourier series?)