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Sinc series

The sinc function is defined as:

    \[\mathrm{sinc} \; x =\left\{\begin{matrix} 1 & , & x=0 \\\\ \dfrac{\sin x}{x} &, & x \neq 0 \end{matrix}\right.\]

Prove that for any couple (\alpha, \beta) of real numbers in (0, 1) the following result holds:

    \[\sum_{n=-\infty}^{\infty} \mathrm{sinc} \; na \; \mathrm{sinc} \; n \beta = \frac{\pi}{\max\{\alpha, \beta \}}\]

Solution

Since

    \[\sum_{n=-\infty}^{\infty} \mathrm{sinc} \; na \; \mathrm{sinc} \; n \beta = 1 + \frac{2}{\alpha \beta} \sum_{n=1}^{\infty} \frac{\sin n \alpha \sin n \beta}{n^2}\]

due to the addition formulas for the sine and cosine functions it is enough to prove the equality

    \[f(\theta) = \sum_{n=1}^{\infty} \frac{\cos n \theta}{n^2} = \frac{\pi^2}{6} - \frac{\theta\left ( 2 \pi - \theta \right )}{4}  \quad \text{forall} \quad \theta \in [0, 2\pi]\]

which is an immediate consequence of the Fourier series by integrating the sawtooth wave function;

    \[\sum_{n=1}^{\infty} \frac{\sin n \theta}{n} = \frac{\pi-\theta}{2}\]

Hence,

    \begin{align*} \sum_{n=-\infty}^{\infty} \mathrm{sinc} \; na \; \mathrm{sinc} \; n \beta &= 1+ \frac{f\left ( \left | \alpha- \beta\right | \right) + f\left ( \alpha+\beta \right )}{\alpha \beta} \\ &=1+\frac{1}{4\alpha \beta} \bigg ( \left ( \alpha-\beta \right )^2 - \left ( \alpha+\beta \right )^2 +\\ &\quad \quad \quad + 2\pi \left ( \alpha-\beta-\left | \alpha+\beta \right | \right ) \bigg ) \\ &= 1+ \frac{1}{4\alpha \beta} \left ( -4\alpha \beta + 4 \pi \min \{ \alpha, \beta \} \right )\\ &= \frac{\pi}{\max\{\alpha, \beta \}} \end{align*}

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